Point a of the circular track. Tasks for circular motion

Consider the movement of two points along a circle of length s in one direction with simultaneous start with speeds v 1 andv 2 (v 1 >v2) and answer the question: after what time will the first point be ahead of the second by exactly one circle? Assuming that the second point is at rest, and the first is approaching it with a speed v 1 -v 2 ., we get that the condition of the problem will be fulfilled when the first point equals the second one for the first time. While the first point will pass the distance, equal to the length one circle, and the desired formula is no different from the formula obtained for the task of moving after:

So, if two points simultaneously start moving along a circle in one direction with speeds v 1 and v 2, respectively (v 1 > v 2, respectively), then the first point approaches the second with a speed v 1 —v2 and at the moment when the first point catches up with the second for the first time, it covers the distance one circle more.

Task 3. From one point circular track, whose length is 14 km, two cars started simultaneously in the same direction. The speed of the first car is 80 km/h, and 40 minutes after the start it was one lap ahead of the second car. Find the speed of the second car. Give your answer in km/h.

Decision. Let the speed of the second car be x km/h. Since 40 minutes is 2/3 of an hour and this is the time for which the first car will be one lap ahead of the second, we will compose the equation according to the condition of the problem

where 160 - 2x \u003d 42, i.e. x \u003d 59.

Answer. 59 km/h

Training tasks

T3.1. From one point of the circular track, the length of which is 15 km, two cars started simultaneously in the same direction. The speed of the first car is 60 km/h, the speed of the second is 80 km/h. How many minutes will pass from the moment of the start before the first car is exactly 1 lap ahead of the second?

T3.2. From one point of the circular track, the length of which is 10 km, two cars started simultaneously in the same direction. The speed of the first car is 90 km/h, and 40 minutes after the start it was one lap ahead of the second car. Find the speed of the second car. Give your answer in km/h.

T3.3. Two motorcycles start simultaneously in the same direction from two diametrically opposite points circular track, the length of which is 20 km. After how many minutes will the motorcycles equalize for the first time if the speed of one of them is 12 km/h more speed another?

T3.4. The clock with hands shows 9 hours 00 minutes. In how many minutes will the minute hand align with the hour hand for the third time?

T3.5. Ski competitions are held on a circular track. The first skier completes one lap 2 minutes faster than the second and an hour later he is exactly one lap ahead of the second. How many minutes does the second skier complete one lap?

T3.6. Two bodies move in a circle in the same direction. The first circle passes 3 minutes faster than the second and catches up with the second every hour and a half. How many minutes does it take the first body to complete one circle?

T3.7. Two points rotate uniformly around a circle. The first makes a revolution 5 seconds faster than the second and makes 2 revolutions more per minute than the second. How many revolutions per minute does the second point make?

T3.8. From point A of the circular track, start simultaneously uniform motion in opposite directions two bodies. At the moment of their meeting, the first body travels 100 meters more than the second, and returns to point A 9 minutes after the meeting. Find the length of the path in meters if the second body returns to point A 16 minutes after the meeting.

x 4x, and the speed of their approach is 3 x km/h.

So 3 X

Answer: 80.

Answer: 75

A cyclist left point A of the circular track, and after 20 minutes a motorcyclist followed him. 5 minutes after departure, he caught up with the cyclist for the first time, and 46 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 46 km. Give your answer in km/h.

Decision.

prototype.

A cyclist left point A of the circular track. After 30 minutes, he had not yet returned to point A, and a motorcyclist followed him from point A. 10 minutes after departure, he caught up with the cyclist for the first time, and 30 minutes after that he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 30 km. Give your answer in km/h.

By the time of the first overtaking, the motorcyclist has traveled as much in 10 minutes as the cyclist in 40 minutes, therefore, his speed is 4 times greater. Therefore, if the speed of the cyclist is taken as x km / h, then the speed of the motorcyclist will be equal to 4x, and the speed of their approach is 3 x km/h.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

Answer: 75

A cyclist left point A of the circular track, and after 40 minutes a motorcyclist followed him. 10 minutes after departure, he caught up with the cyclist for the first time, and 47 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 47 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

A cyclist left point A of the circular track, and after 40 minutes a motorcyclist followed him. 10 minutes after departure, he caught up with the cyclist for the first time, and 36 minutes after that he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 36 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

Answer: 75

A cyclist left point A of the circular track, and after 40 minutes a motorcyclist followed him. 8 minutes after departure, he caught up with the cyclist for the first time, and 21 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 35 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

A cyclist left point A of the circular track, and after 20 minutes a motorcyclist followed him. 5 minutes after departure, he caught up with the cyclist for the first time, and 8 minutes after that he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 8 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

Answer:

A cyclist left point A of the circular track, and after 40 minutes a motorcyclist followed him. 5 minutes after departure, he caught up with the cyclist for the first time, and 9 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 12 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

A cyclist left point A of the circular track, and after 30 minutes a motorcyclist followed him. 10 minutes after departure, he caught up with the cyclist for the first time, and 40 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 40 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

Answer: 0

A cyclist left point A of the circular track, and after 30 minutes a motorcyclist followed him. 4 minutes after departure, he caught up with the cyclist for the first time, and 32 minutes after that he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 40 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

Answer: 80

A cyclist left point A of the circular track, and after 50 minutes a motorcyclist followed him. 10 minutes after departure, he caught up with the cyclist for the first time, and 36 minutes after that he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 45 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

Answer: 80

A cyclist left point A of the circular track, and after 20 minutes a motorcyclist followed him. 5 minutes after departure, he caught up with the cyclist for the first time, and 40 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 40 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

Answer: 1

A cyclist left point A of the circular track, and after 40 minutes a motorcyclist followed him. 25 minutes after departure, he caught up with the cyclist for the first time, and 39 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 26 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

Answer: 65

A cyclist left point A of the circular track, and after 30 minutes a motorcyclist followed him. 15 minutes after departure, he caught up with the cyclist for the first time, and 42 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 35 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

Answer: 1

A cyclist left point A of the circular track, and after 20 minutes a motorcyclist followed him. 8 minutes after departure, he caught up with the cyclist for the first time, and 54 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 45 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

Answer: 70

A cyclist left point A of the circular track, and after 50 minutes a motorcyclist followed him. 10 minutes after departure, he caught up with the cyclist for the first time, and 24 minutes after that he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 30 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

A cyclist left point A of the circular track, and after 30 minutes a motorcyclist followed him. 5 minutes after departure, he caught up with the cyclist for the first time, and 47 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 47 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

Answer: 70

A cyclist left point A of the circular track, and after 10 minutes a motorcyclist followed him. 5 minutes after departure, he caught up with the cyclist for the first time, and 51 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 34 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

A cyclist left point A of the circular track, and after 50 minutes a motorcyclist followed him. 5 minutes after departure, he caught up with the cyclist for the first time, and 9 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 15 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

A cyclist left point A of the circular track, and after 30 minutes a motorcyclist followed him. 8 minutes after departure, he caught up with the cyclist for the first time, and 12 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 15 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

A cyclist left point A of the circular track, and after 40 minutes a motorcyclist followed him. 12 minutes after departure, he caught up with the cyclist for the first time, and 54 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 45 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

A cyclist left point A of the circular track, and after 50 minutes a motorcyclist followed him. 10 minutes after departure, he caught up with the cyclist for the first time, and 18 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 30 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

Answer:

A cyclist left point A of the circular track, and after 40 minutes a motorcyclist followed him. 16 minutes after departure, he caught up with the cyclist for the first time, and 24 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 20 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

A cyclist left point A of the circular track, and after 50 minutes a motorcyclist followed him. 25 minutes after departure, he caught up with the cyclist for the first time, and 30 minutes after that he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 25 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

Answer: 150

A cyclist left point A of the circular track, and after 50 minutes a motorcyclist followed him. 25 minutes after departure, he caught up with the cyclist for the first time, and 54 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 36 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

A cyclist left point A of the circular track, and after 40 minutes a motorcyclist followed him. 8 minutes after departure, he caught up with the cyclist for the first time, and 28 minutes after that he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 35 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

A cyclist left point A of the circular track, and after 30 minutes a motorcyclist followed him. 10 minutes after departure, he caught up with the cyclist for the first time, and 48 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 36 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

Answer: 1

A cyclist left point A of the circular track, and after 20 minutes a motorcyclist followed him. 2 minutes after departure, he caught up with the cyclist for the first time, and 9 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 15 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

A cyclist left point A of the circular track, and after 40 minutes a motorcyclist followed him. 25 minutes after departure, he caught up with the cyclist for the first time, and 57 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 38 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

A cyclist left point A of the circular track, and after 50 minutes a motorcyclist followed him. 10 minutes after departure, he caught up with the cyclist for the first time, and 12 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 15 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

A cyclist left point A of the circular track, and after 20 minutes a motorcyclist followed him. 6 minutes after departure, he caught up with the cyclist for the first time, and 12 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 10 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

A cyclist left point A of the circular track, and after 30 minutes a motorcyclist followed him. 12 minutes after departure, he caught up with the cyclist for the first time, and 48 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 40 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

A cyclist left point A of the circular track, and after 10 minutes a motorcyclist followed him. 1 minute after departure, he caught up with the cyclist for the first time, and 15 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 25 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

A cyclist left point A of the circular track, and after 30 minutes a motorcyclist followed him. 6 minutes after departure, he caught up with the cyclist for the first time, and 16 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 20 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

A cyclist left point A of the circular track, and after 40 minutes a motorcyclist followed him. 12 minutes after departure, he caught up with the cyclist for the first time, and 18 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 15 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

A cyclist left point A of the circular track, and after 30 minutes a motorcyclist followed him. 10 minutes after departure, he caught up with the cyclist for the first time, and 21 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 21 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

A cyclist left point A of the circular track, and after 10 minutes a motorcyclist followed him. 2 minutes after departure, he caught up with the cyclist for the first time, and 28 minutes after that he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 35 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

Answer: .

A cyclist left point A of the circular track, and after 20 minutes a motorcyclist followed him. 5 minutes after departure, he caught up with the cyclist for the first time, and 27 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 27 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

A cyclist left point A of the circular track, and after 40 minutes a motorcyclist followed him. 10 minutes after departure, he caught up with the cyclist for the first time, and 19 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 19 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

A cyclist left point A of the circular track, and after 40 minutes a motorcyclist followed him. 20 minutes after departure, he caught up with the cyclist for the first time, and 36 minutes after that he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 24 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

A cyclist left point A of the circular track, and after 40 minutes a motorcyclist followed him. 4 minutes after departure, he caught up with the cyclist for the first time, and 12 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 20 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

Answer: 110

A cyclist left point A of the circular track, and after 20 minutes a motorcyclist followed him. 4 minutes after departure, he caught up with the cyclist for the first time, and 32 minutes after that he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 40 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

A cyclist left point A of the circular track, and after 20 minutes a motorcyclist followed him. 5 minutes after departure, he caught up with the cyclist for the first time, and 48 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 48 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

A cyclist left point A of the circular track, and after 30 minutes a motorcyclist followed him. 10 minutes after departure, he caught up with the cyclist for the first time, and 56 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 42 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

A cyclist left point A of the circular track, and after 50 minutes a motorcyclist followed him. 25 minutes after departure, he caught up with the cyclist for the first time, and 36 minutes after that he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 30 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

A cyclist left point A of the circular track, and after 40 minutes a motorcyclist followed him. 5 minutes after departure, he caught up with the cyclist for the first time, and 36 minutes after that he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 48 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

A cyclist left point A of the circular track, and after 30 minutes a motorcyclist followed him. 3 minutes after departure, he caught up with the cyclist for the first time, and 9 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 15 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

A cyclist left point A of the circular track, and after 40 minutes a motorcyclist followed him. 10 minutes after departure, he caught up with the cyclist for the first time, and 25 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 25 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

A cyclist left point A of the circular track, and after 30 minutes a motorcyclist followed him. 15 minutes after departure, he caught up with the cyclist for the first time, and 57 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 38 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

A cyclist left point A of the circular track, and after 40 minutes a motorcyclist followed him. 8 minutes after departure, he caught up with the cyclist for the first time, and 32 minutes after that he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 40 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

A cyclist left point A of the circular track, and after 30 minutes a motorcyclist followed him. 5 minutes after departure, he caught up with the cyclist for the first time, and 12 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 18 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

A cyclist left point A of the circular track, and after 40 minutes a motorcyclist followed him. 8 minutes after departure, he caught up with the cyclist for the first time, and 54 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 45 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

A cyclist left point A of the circular track, and after 40 minutes a motorcyclist followed him. 10 minutes after departure, he caught up with the cyclist for the first time, and 43 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 43 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

So 3 X\u003d 60 km / h, from where the speed of the cyclist is 20 km / h, and the speed of the motorcyclist is 80 km / h.

Answer: 80.

A cyclist left point A of the circular track, and after 20 minutes a motorcyclist followed him. 4 minutes after departure, he caught up with the cyclist for the first time, and 21 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 35 km. Give your answer in km/h.

Decision.

This task has not yet been solved, we present the solution of the prototype.

On the other hand, the second time the motorcyclist caught up with the cyclist in 30 minutes, during this time he traveled 30 km more. Therefore, the speed of their convergence will be 60 km/h.

Type of lesson: iterative-generalizing lesson.

Lesson Objectives:

educational

various types

word problems

developing

educational

educational process

Lesson equipment:

interactive board;
envelopes with tasks, thematic control cards, consultant cards.

Lesson structure.

	The main stages of the lesson	Tasks to be solved at this stage
	Organizing time, introductory part	creation friendly atmosphere in class set students up for productive work identify missing check students readiness for the lesson
	Preparing students for active work(repetition)	check students' knowledge on the topic: "Solving text problems of various types for movement" implementation of the development of speech and thinking of responding students development of analytical and critical thinking of students through commenting on the answers of classmates organize learning activities of the whole class during the response of the students called to the board
	The stage of generalization and systematization of the studied material (work in groups)	test students' ability to solve problems of various types of movement, to form students' knowledge reflected in the form of ideas and theories, the transition from private ideas to broader generalizations to carry out the formation of moral relations of students to participants in the educational process (during group work)
	Examination work, adjustment (if necessary)	check the execution of data for groups of tasks (their correctness) continue to form students' ability to analyze, highlight the main thing, build analogies, generalize and systematize develop the ability to negotiate
	Summing up the lesson. Parsing homework	inform students about homework, explain the methodology for its implementation motivate the need and obligation to do homework sum up the lesson

Forms of organization cognitive activity students:

frontal form of cognitive activity - at stages II, IY, Y.
group form of cognitive activity - at stage III.

Teaching methods: verbal, visual, practical, explanatory - illustrative, reproductive, partially - search, analytical, comparative, generalizing, traductive.

During the classes

I. Organizational moment, introductory part.

The teacher announces the topic of the lesson, the objectives of the lesson and the main points of the lesson. Checks the readiness of the class to work.

II. Preparing students for active work (review)

Answer the questions.

What kind of movement is called uniform (movement at a constant speed).
What is the path formula for uniform motion ( S=Vt).
From this formula, express the speed and time.
Specify units of measure.

Conversion of speed units

III. The stage of generalization and systematization of the studied material (work in groups)

The whole class is divided into groups (5-6 people in a group). It is desirable that there are students in the same group different levels preparation. Among them, a group leader (the strongest student) is appointed, who will lead the work of the group.

All groups receive envelopes with assignments (they are the same for all groups), consultant cards (for weak students) and thematic control sheets. In sheets thematic control the group leader assigns marks to each student of the group for each task and notes the difficulties that students have in completing specific tasks.

Card with tasks for each group.

№ 5.

No. 7. The motorboat passed 112 km against the current of the river and returned to the point of departure, having spent 6 hours less on the way back. Find the speed of the current if the speed of the boat in still water is 11 km/h. Give your answer in km/h.

No. 8. The motor ship passes along the river to the destination 513 km and after parking returns to the point of departure. Find the speed of the ship in still water, if the speed of the current is 4 km/h, the stay lasts 8 hours, and the ship returns to the point of departure 54 hours after leaving it. Give your answer in km/h.

No. 9. From pier A to pier B, the distance between which is 168 km, the first ship set off at a constant speed, and 2 hours after that, the second one set off after it, at a speed of 2 km / h more. Find the speed of the first ship if both ships arrive at point B at the same time. Give your answer in km/h.

Sample of thematic control card.

Class ________ Full name of the student ___________________________________

job number		Comment

Consultant cards.

Card number 1 (consultant)

1. Driving on a straight road

When solving problems of uniform motion, two situations often occur.

If a initial distance between objects is equal to S , and the speeds of objects V1 and V2, then:

a) when objects move towards each other, the time after which they will meet is equal to .

b) when objects move in one side time, through which the first object will catch up with the second, equals , ( V 2 > V 1)

Example 1. The train, having traveled 450 km, was stopped due to snow drift. Half an hour later the path was cleared, and the driver, having increased the speed of the train by 15 km/h, brought it to the station without delay. Find the initial speed of the train if the distance traveled by it to the stop was 75% of the total distance.

Find the whole path: 450: 0.75 = 600 (km)
Let's find the length of the second section: 600 - 450 = 150 (km)
Let's make and solve the equation:

X= -75 is not suitable for the condition of the problem, where x > 0.

Answer: The initial speed of the train is 60 km/h.

Card number 2 (consultant)

2. Driving on a closed road

If the length of the closed road is S, and the speeds of objects V 1 and V 2 , then:

a) when objects move in different directions the time between their meetings is calculated by the formula ;
b) when objects move in one direction, the time between their meetings is calculated by the formula

Example 2 At competitions on the ring track, one skier completes the circle 2 minutes faster than the other and after an hour has bypassed him exactly on the circle. How long does it take each skier to complete the lap?

Let be S m is the length of the ring road and x m/min and y m/min are the speeds of the first and second skiers, respectively ( x > y) .

Then S/x min and S/y min - the time for which the first and second skiers pass the circle, respectively. From the first condition we obtain the equation . Since the speed of removal of the first skier from the second skier is ( x- y) m/min, then from the second condition we have the equation .

Let's solve the system of equations.

Let's make a replacement S/x=a and S/y=b, then the system of equations will take the form:

. Multiply both sides of the equation by 60 a(a + 2) > 0.

60(a + 2) – 60a = a(a + 2)a 2 + 2a- 120 = 0. Quadratic equation has one positive root a = 10 then b= 12. So the first skier completes the lap in 10 minutes, and the second skier in 12 minutes.

Answer: 10 min; 12 min.

Card number 3 (consultant)

3. Movement on the river

If an object moves along the river, then its speed is equal to Vstream. =Voct. + Vtech.

If an object is moving against the current of the river, then its speed is Vagainst the current =V oct. – Vtech. The object’s own speed (speed in still water) is equal to

The speed of the river is

The speed of the raft is equal to the speed of the river.

Example 3 The boat went downstream for 50 km and then traveled 36 km in the opposite direction, which took him 30 minutes longer than downstream. What is the speed of the boat if the speed of the river is 4 km/h?

Let the boat's own speed be X km/h, then its speed along the river is ( x + 4) km / h, and against the current of the river ( x- 4) km/h. The time of the boat's movement along the river is equal to hours, and against the flow of the river, hours. Since 30 minutes = 1/2 hour, then, according to the condition of the problem, we will compose the equation =. Multiply both sides of the equation by 2( x + 4)(x- 4) >0 .

We get 72( x + 4) -100(x- 4) = (x + 4)(x- 4) x 2 + 28x- 704 \u003d 0 x 1 \u003d 16, x 2 \u003d - 44 (we exclude, since x> 0).

So, the own speed of the boat is 16 km/h.

Answer: 16 km/h.

IV. Problem solving stage.

Problems that caused difficulties for students are analyzed.

No. 1. From two cities, the distance between which is equal to 480 km, two cars simultaneously left towards each other. In how many hours will the cars meet if their speeds are 75 km/h and 85 km/h?

75 + 85 = 160 (km/h) – closing speed.
480: 160 = 3 (h).

Answer: the cars will meet in 3 hours.

No. 2. From cities A and B, the distance between them is 330 km, two cars simultaneously left towards each other and met after 3 hours at a distance of 180 km from city B. Find the speed of the car that left city A. Give your answer in km / h.

(330 - 180) : 3 = 50 (km/h)

Answer: The speed of a car leaving city A is 50 km/h.

No. 3. From point A to point B, the distance between which is 50 km, a motorist and a cyclist left at the same time. It is known that a motorist travels 65 km more per hour than a cyclist. Determine the speed of the cyclist if it is known that he arrived at point B 4 hours 20 minutes later than the motorist. Give your answer in km/h.

Let's make a table.

Let's make an equation, given that 4 hours 20 minutes =

It is obvious that x = -75 does not fit the condition of the problem.

Answer: The speed of the cyclist is 10 km/h.

No. 4. Two motorcyclists start simultaneously in one direction from two diametrically opposite points of a circular track, the length of which is 14 km. In how many minutes will the motorcyclists catch up for the first time if the speed of one of them is 21 km/h more than the speed of the other?

Let's make a table.

Let's make an equation.

where 1/3 hour = 20 minutes.

Answer: After 20 minutes, the motorcyclists will line up for the first time.

No. 5. From one point of the circular track, the length of which is 12 km, two cars started simultaneously in the same direction. The speed of the first car is 101 km/h, and 20 minutes after the start it was one lap ahead of the second car. Find the speed of the second car. Give your answer in km/h.

Let's make a table.

Let's make an equation.

Answer: The speed of the second car is 65 km/h.

No. 6. A cyclist left point A of the circular track, and after 40 minutes a motorcyclist followed him. 8 minutes after departure, he caught up with the cyclist for the first time, and 36 minutes after that he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 30 km. Give your answer in km/h.

Let's make a table.

	Movement to the first meeting

cyclist

We continue to consider tasks for movement. There is a group of tasks that differs from the usual tasks for movement - these are tasks for Roundabout Circulation(circular track, clock movement). In this article, we will consider such tasks. The principles of the solution are the same, the same (the formula for the law of rectilinear motion). But there are small nuances in the approaches to the solution.

Consider the tasks:

Two motorcyclists start simultaneously in the same direction from two diametrically opposite points of a circular track, the length of which is 22 km. In how many minutes will the motorcyclists catch up for the first time if the speed of one of them is 20 km/h more than the speed of the other?

At first glance, some people may find roundabout tasks difficult and somewhat confusing compared to ordinary tasks on rectilinear motion. But this is only at first glance. This problem easily turns into a problem of rectilinear motion. How?

Mentally turn the circular track into a straight line. There are two motorcyclists on it. One of them is 11 km behind the other, as it is stated in the condition that the length of the track is 22 kilometers.

The speed of the lagging behind is 20 kilometers per hour more (he catches up with the one who is ahead). Here is the problem for rectilinear motion.

So, the desired value (the time after which they become equal) will be taken as x hours. The speed of the first one (the one in front) will be denoted by y km/h, then the speed of the second one (the overtaking one) will be y + 20.

Let's put the speed and time in the table.

Fill in the column "distance":

The second travels a distance (to a meeting) 11 km more, which means

11/20 hours is the same as 33/60 hours. That is, 33 minutes had passed before their meeting. How to convert hours to minutes, and vice versa, you can see in the article "".

As you can see, the very speed of motorcyclists in this case irrelevant.

Answer: 33

Decide for yourself:

Two motorcyclists start simultaneously in the same direction from two diametrically opposite points of a circular track, the length of which is 14 km. In how many minutes will the motorcyclists catch up for the first time if the speed of one of them is 21 km/h more than the speed of the other?

From one point of the circular track, the length of which is 25 km, two cars started simultaneously in the same direction. The speed of the first car is 112 km/h, and 25 minutes after the start it was one lap ahead of the second car. Find the speed of the second car. Give your answer in km/h.

This problem can also be interpreted, that is, presented as a problem for rectilinear motion. How? Just …

Two cars start moving in the same direction at the same time. The speed of the first is 112 km/h. After 25 minutes, he is ahead of the second by 25 km (since it is said that by one lap). Find the speed of the second. It is very important to represent the process of this movement in the problems of movement.

We will make a comparison by distance, since we know that one was ahead of the other by 25 kilometers.

For x we take the desired value - the speed of the second. Travel time 25 minutes (25/60 hours) for both.

Fill in the column "distance":

The distance traveled by the first is 25 km more than the distance traveled by the second. I.e:

The speed of the second car is 52 (km/h).

Answer: 52

Decide for yourself:

From one point of the circular track, the length of which is 14 km, two cars started simultaneously in the same direction. The speed of the first car is 80 km/h, and 40 minutes after the start it was one lap ahead of the second car. Find the speed of the second car. Give your answer in km/h.

A cyclist left point A of the circular track, and after 40 minutes a motorcyclist followed him. 8 minutes after departure, he caught up with the cyclist for the first time, and 36 minutes after that he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 30 km. Give your answer in km/h.

This task is relatively difficult. What is immediately worth noting? This is that a motorcyclist travels the same distance as a cyclist, catching up with him for the first time. Then he again catches up with him for the second time, and the difference in distances covered after the first meeting is 30 kilometers (circle length). Thus, it will be possible to compose two equations and solve their system. We are not given the speed of the participants in the movement, so it will be possible to introduce two variables. A system of two equations with two variables is solved.

So, let's convert minutes to hours, since the speed must be found in km / h.

Forty minutes is 2/3 of an hour, 8 minutes is 8/60 of an hour, 36 minutes is 36/60 of an hour.

The speeds of the participants will be denoted as x km/h (for a cyclist) and y km/h (for a motorcyclist).

For the first time, the motorcyclist overtook the cyclist after 8 minutes, that is, 8/60 hours after the start.

Up to this point, the cyclist has been on the road for 40 + 8 = 48 minutes, that is, 48/60 hours.

Let's write this data in a table:

Both have traveled the same distance, that is

Then the motorcyclist caught up with the cyclist a second time. This happened after 36 minutes, that is, 36/60 hours after the first overtaking.

Let's make a second table, fill in the column "distance":

Since it is said that after 36 minutes the motorcyclist caught up with the cyclist again. This means that he (the motorcyclist) traveled a distance equal to 30 kilometers (one lap) plus the distance that the cyclist traveled during this time. This is key moment for the second equation.

One circle is the length of the track, it is equal to 30 km.

We get the second equation:

We solve a system of their two equations:

So y \u003d 6 ∙ 10 \u003d 60.

That is, the speed of the motorcyclist is 60 km/h.

Answer: 60

Decide for yourself:

A cyclist left point A of the circular track, and after 30 minutes a motorcyclist followed him. 10 minutes after departure, he caught up with the cyclist for the first time, and 30 minutes after that he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 30 km. Give your answer in km/h.

The next type of circular motion problems can be said to be “unique”. There are tasks that are solved orally. And there are those that are extremely difficult to solve without understanding and attentiveness in reasoning. We are talking about tasks about the hands of the clock.

Here is an example of a simple task:

The clock with hands shows 11 hours 20 minutes. After how many minutes does the minute hand equalize with the hour hand for the first time?

The answer is obvious, in 40 minutes, when it will be exactly twelve. Even if they could not immediately understand, then by drawing the dial(making a sketch) on the sheet, you can easily determine the answer.

Examples of other tasks (not easy):

The clock with hands shows 6 hours 35 minutes. After how many minutes will the minute hand align with the hour hand for the fifth time? Answer: 325

The clock with hands shows exactly 2 o'clock. In how many minutes will the minute hand align with the hour hand for the tenth time? Answer: 600

Decide on your own:

The clock with hands shows 8 hours 00 minutes. After how many minutes will the minute hand align with the hour hand for the fourth time?

Are you convinced that it is very easy to get confused?

In general, I am not a supporter of giving such advice, but here it is needed, since on the exam you can easily get confused with such a task, calculate incorrectly, or simply lose a lot of time on solving.

you can decide this task in one minute. How? Just!

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That's all. I wish you success!

Sincerely, Alexander.

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1. Two cars left point A for point B at the same time. The first traveled at a constant speed all the way. The second car traveled the first half of the way at a speed less than the speed of the first by 15 km/h, and the second half of the way at a speed of 90 km/h, as a result of which it arrived at point B at the same time as the first car. Find the speed of the first car if it is known to be greater than 54 km/h. Give your answer in km/h.

2. A train moving uniformly at a speed of 60 km/h passes a forest belt 400 meters long in 1 minute. Find the length of the train in meters.

3. The distance between cities A and B is 435 km. The first car drove from city A to city B at a speed of 60 km/h, and an hour later, a second car drove towards it at a speed of 65 km/h. At what distance from city A will the cars meet? Give your answer in kilometers.

4. On two parallel railway tracks a freight train and a passenger train follow in the same direction, the speeds of which are respectively 40 km/h and 100 km/h. The length of a freight train is 750 m. Find the length of a passenger train if the time it takes for it to pass the freight train is 1 minute.

5. A train moving uniformly at a speed of 63 km/h passes a pedestrian walking in the same direction parallel to the tracks at a speed of 3 km/h in 57 seconds. Find the length of the train in meters.

6. Solving motion problems.

7. The road between points A and B consists of an ascent and a descent, and its length is 8 km. A pedestrian traveled from A to B in 2 hours and 45 minutes. The time of its movement on the descent was 1 hour 15 minutes. With what speed did the pedestrian walk on the downhill if the speed of his movement on the ascent is less than the speed of the movement on the descent by 2 km / h. Express your answer in km/h.

8. The car drove from the city to the village in 3 hours. If he increased his speed by 25 km/h, he would spend 1 hour less on this journey. How many kilometers is the distance from the city to the village?

http://youtu.be/x64JkS0XcrU

9. Ski competitions are held on a circular track. The first skier completes one lap 2 minutes faster than the second and an hour later he is exactly one lap ahead of the second. How many minutes does the second skier complete one lap?

10. Two motorcyclists start simultaneously in the same direction from two diametrically opposite points of a circular track, the length of which is 6 km. In how many minutes will the motorcyclists catch up for the first time if the speed of one of them is 18 km/h more than the speed of the other?

Movement problems from Anna Denisova. Site http://easy-physic.ru/

11. Video lecture. 11 tasks for movement.

1. A cyclist travels 500 m less every minute than a motorcyclist, so he spends 2 hours more on a 120 km journey. Find the speeds of the cyclist and motorcyclist.

2. The motorcyclist stopped for refueling for 12 minutes. After that, increasing the speed by 15 km / h, he made up Lost time at a distance of 60 km. How fast did he move after stopping?

3. Two motorcyclists set off simultaneously towards each other from points A and B, the distance between which is 600 km. While the first travels 250 km, the second manages to overcome 200 km. Find the velocities of the motorcyclists if the first one arrives at B three hours earlier than the second at A.

4. The plane was flying at a speed of 220 km/h. When he had to fly 385 km less than he had already overcome, the plane increased its speed to 330 km / h. The average speed of the aircraft for the entire journey was 250 km/h. How far has the plane traveled before increasing speed?

5. By railway the distance from A to B is 88 km, by water it increases to 108 km. The train from A leaves 1 hour later than the ship and arrives at B 15 minutes earlier. Find the average speed of the train, if it is known that it is 40 km/h more than the average speed of the ship.

6. Two cyclists have left two places 270 km apart and are riding towards each other. The second one travels 1.5 km less per hour than the first one, and meets him in as many hours as the first one does in kilometers per hour. Determine the speed of each cyclist.

7. Two trains depart from points A and B towards each other. If the trains from A leave two hours earlier than the train from B, they will meet half way. If they leave at the same time, then in two hours the distance between them will be 0.25 of the distance between points A and B. How many hours does each train take to complete the journey?

8. The train passed a person standing motionless on the platform in 6 seconds, and past a platform 150 m long - in 15 seconds. Find the speed of the train and its length.

9. A train 1 km long passed the pole in 1 minute, and through the tunnel (from the entrance of the locomotive to the exit of the last car) at the same speed - in 3 minutes. What is the length of the tunnel (in km)?

10. From stations A and B, the distance between which is 75 km, freight and fast trains set off at the same time, and met in half an hour. The freight train arrived at B 25 minutes later than the fast train at A. What is the speed of each train?

11. Piers A and B are located on the river, the speed of which in this section is 4 km / h. The boat goes from A to B and back without stopping from average speed 6 km/h Find own speed boats.

12. Video lecture. 8 tasks for moving in a circle

12. Two points move uniformly and in the same direction along a circle 60 m long. One of them does full turn 5 seconds faster than another. In this case, the coincidence of points occurs every time after 1 minute. Find the velocities of the points.

13. How much time elapses between two successive coincidences of the hour and minute hands on a watch dial?

14. Two runners start from one point of the stadium ring track, and the third one - from a diametrically opposite point simultaneously with them in the same direction. After running three laps, the third runner caught up with the second. Two and a half minutes later, the first runner caught up with the third. How many laps per minute does the second runner run if the first overtakes him once every 6 minutes?

15. Three riders start at the same time from the same point on the track, which has the shape of a circle and ride in the same direction with constant speeds. The first rider caught up with the second for the first time, making his fifth lap, at a point diametrically opposite to the start, and half an hour after that, he overtook the third rider for the second time, not counting the start moment. The second rider caught up with the third for the first time three hours after the start. How many laps per hour does the first rider make if the second rider completes the lap in at least 20 minutes?

16. Two motorcyclists start simultaneously in the same direction from two diametrically opposite points of a circular track, the length of which is 14 km. In how many minutes will the motorcyclists catch up for the first time if the speed of one of them is 21 km/h more than the speed of the other?

17. A cyclist left point A of the circular track, and after 30 minutes a motorcyclist followed him. In 10 minutes. after departure, he caught up with the cyclist for the first time, and 30 minutes later he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 30 km. Give your answer in km/h.

18. Clock with hands show 3 o'clock exactly. After how many minutes will the minute hand align with the hour hand for the ninth time?

18.1 Two riders are racing. They have to drive 60 laps on a 3 km long ring road. Both riders started at the same time, and the first one came to the finish line 10 minutes earlier than the second one. What was the average speed of the second rider if it is known that the first rider overtook the second for the first time in a lap in 15 minutes?

13. Video lecture. 6 tasks for movement on water.

19. Cities A and B are located on the banks of the river, with city B downstream. At 9 o'clock in the morning, a raft departs from city A to city B. At the same moment, a boat leaves from B to A, which meets the raft in 5 hours. Having sailed to city A, the boat turns back and sails to B at the same time as the raft. Will the boat and raft arrive in city B by nine o'clock that evening?

20. A motor boat left point A for point B against the current of the river. On the way, the motor broke down, and while it was being repaired for 20 minutes, the boat was demolished down the river. Determine how late the boat arrived at point B if usually the journey from A to B takes one and a half times longer than from B to A?

21. Cities A and B are located on the banks of the river, with city A downstream. Two boats leave these cities at the same time towards each other, which meet in the middle between the cities. After the meeting, the boats continue their journey, and, having reached cities A and B, respectively, turn around and meet again at a distance of 20 km from the place of the first meeting. If the boats had initially swum against the current, then the boat leaving A would have overtaken the boat leaving B 150 km from B. Find the distance between the cities.

22. Two steamers, the speed of which is standing water the same, depart from two piers: the first from A downstream, the second from B upstream. Each ship stops at its destination for 45 minutes and returns back. If the steamboats depart simultaneously from the starting points, then they meet at point K, which is two times closer to A than to B. If the first steamer departs from A 1 hour later than the second departs from B, then way back steamers meet 20 km from A. If the first steamer leaves from A 30 minutes earlier than the second from B, then on the way back they meet 5 km above K. Find the speed of the river and the time it takes the second steamer to reach A to K.

23. From point A to point B, located downstream of the river, a raft set off. At the same time, a boat left point B to meet him. Having met the raft, the boat immediately turned and swam back. What part of the way from A to B will the raft cover by the time the boat returns to point B, if the speed of the boat in still water is four times the speed of the current?

24. Piers A and B are located on the river, the speed of which in this section is 4 km / h. The boat travels from A to B and back at an average speed of 6 km/h. Find your boat's own speed.

Portal for the student. Self-training

During the classes

I. Organizational moment, introductory part.

II. Preparing students for active work (review)

III. The stage of generalization and systematization of the studied material (work in groups)

IV. Problem solving stage.

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