This material is a system of tasks on the topic “Movement”.

Purpose: to help students more fully master the technologies for solving problems on this topic.

Tasks for movement on water.

Very often a person has to make movements on water: river, lake, sea.

At first he did it himself, then rafts, boats, sailing ships appeared. With the development of technology, steamships, motor ships, nuclear-powered ships came to the aid of man. And he was always interested in the length of the path and the time spent on overcoming it.

Imagine that it is spring outside. The sun melted the snow. Puddles appeared and streams ran. Let's make two paper boats and put one of them into a puddle, and the second into a stream. What will happen to each of the ships?

In a puddle, the boat will stand still, and in a stream it will float, as the water in it "runs" to a lower place and carries it with it. The same will happen with a raft or a boat.

In the lake they will stand still, and in the river they will swim.

Consider the first option: a puddle and a lake. Water does not move in them and is called standing.

The boat will float in a puddle only if we push it or if the wind blows. And the boat will begin to move in the lake with the help of oars or if it is equipped with a motor, that is, due to its speed. Such a movement is called movement in still water.

Is it different from driving on the road? Answer: no. And this means that we know how to act in this case.

Problem 1. The speed of the boat on the lake is 16 km/h.

How far will the boat travel in 3 hours?

Answer: 48 km.

It should be remembered that the speed of a boat in still water is called own speed.

Problem 2. A motorboat sailed 60 km across the lake in 4 hours.

Find the own speed of the motorboat.

Answer: 15 km/h.

Task 3. How long will it take for a boat whose own speed is

equals 28 km/h to swim 84 km across the lake?

Answer: 3 hours.

So, To find the distance traveled, you need to multiply the speed by the time.

To find the speed, you need to divide the distance by the time.

To find the time, you need to divide the distance by the speed.

What is the difference between driving on a lake and driving on a river?

Recall a paper boat in a stream. It floated because the water in it moves.

Such a movement is called downstream. And in the opposite direction - moving against the current.

So, the water in the river moves, which means it has its own speed. And they call her river speed. (How to measure it?)

Problem 4. The speed of the river is 2 km/h. How many kilometers does the river

any object (wood chip, raft, boat) in 1 hour, in 4 hours?

Answer: 2 km/h, 8 km/h.

Each of you swam in the river and remembers that it is much easier to swim with the current than against the current. Why? Because in one direction the river "helps" to swim, and in the other it "hinders".

Those who do not know how to swim can imagine a situation where a strong wind is blowing. Consider two cases:

1) the wind blows in the back,

2) the wind blows in the face.

In both cases it is difficult to go. The wind in the back makes us run, which means that the speed of our movement increases. The wind in the face knocks us down, slows down. The speed is thus reduced.

Let's take a look at the flow of the river. We have already talked about the paper boat in the spring stream. The water will carry it along with it. And the boat, launched into the water, will float with the speed of the current. But if she has her own speed, then she will swim even faster.

Therefore, in order to find the speed of movement along the river, it is necessary to add the own speed of the boat and the speed of the current.

Problem 5. The own speed of the boat is 21 km/h, and the speed of the river is 4 km/h. Find the speed of the boat along the river.

Answer: 25km/h.

Now imagine that the boat has to sail against the current of the river. Without a motor, or at least an oar, the current would carry her in the opposite direction. But, if you give the boat its own speed (start the engine or land a rower), the current will continue to push it back and prevent it from moving forward at its own speed.

That's why to find the speed of the boat against the current, it is necessary to subtract the speed of the current from its own speed.

Problem 6. The speed of the river is 3 km/h, and the own speed of the boat is 17 km/h.

Find the speed of the boat against the current.

Answer: 14 km/h.

Problem 7. The own speed of the ship is 47.2 km/h, and the speed of the river is 4.7 km/h. Find the speed of the boat upstream and downstream.

Answer: 51.9 km / h; 42.5 km/h.

Problem 8. The speed of a motor boat downstream is 12.4 km/h. Find the own speed of the boat if the speed of the river is 2.8 km/h.

Answer: 9.6 km/h.

Problem 9. The speed of the boat against the current is 10.6 km/h. Find the boat's own speed and the speed with the current if the speed of the river is 2.7 km/h.

Answer: 13.3 km/h; 16 km/h

Relationship between downstream and upstream speed.

Let us introduce the following notation:

V s. - own speed,

V tech. - flow speed,

V on current - flow speed,

V pr.tech. - speed against the current.

Then the following formulas can be written:

V no tech = V c + V tech;

V n.p. flow = V c - V flow;

Let's try to represent it graphically:

Conclusion: the difference in velocities downstream and upstream is equal to twice the current velocity.

Vno tech - Vnp. tech = 2 Vtech.

Vtech \u003d (V by tech - Vnp. tech): 2

1) The speed of the boat upstream is 23 km/h and the speed of the current is 4 km/h.

Find the speed of the boat with the current.

Answer: 31 km/h.

2) The speed of a motorboat downstream is 14 km/h/ and the speed of the current is 3 km/h. Find the speed of the boat against the current

Answer: 8 km/h.

Task 10. Determine the speeds and fill in the table:

* - when solving item 6, see Fig. 2.

Answer: 1) 15 and 9; 2) 2 and 21; 3) 4 and 28; 4) 13 and 9; 5) 23 and 28; 6) 38 and 4.

Solving problems on "movement on water" is difficult for many. There are several types of speeds in them, so the decisive ones start to get confused. To learn how to solve problems of this type, you need to know the definitions and formulas. The ability to draw up diagrams greatly facilitates the understanding of the problem, contributes to the correct compilation of the equation. A correctly composed equation is the most important thing in solving any type of problem.

Instruction

In the tasks "on the movement along the river" there are speeds: own speed (Vс), speed with the flow (Vflow), speed against the current (Vpr.flow), current speed (Vflow). It should be noted that the own speed of a watercraft is the speed in still water. To find the speed with the current, you need to add your own to the speed of the current. In order to find the speed against the current, it is necessary to subtract the speed of the current from the own speed.

The first thing you need to learn and know "by heart" is the formulas. Write down and remember:

Vac = Vc + Vac

Vpr. tech.=Vs-Vtech.

Vpr. flow = Vac. - 2Vtech.

Vac.=Vpr. tech+2Vtech

Vtech.=(Vstream. - Vpr.tech.)/2

Vc=(Vac.+Vc.flow)/2 or Vc=Vac.+Vc.

Using an example, we will analyze how to find your own speed and solve problems of this type.

Example 1. Boat speed downstream is 21.8 km/h and upstream is 17.2 km/h. Find your own speed of the boat and the speed of the river.

Solution: According to the formulas: Vc \u003d (Vac. + Vpr.ch.) / 2 and Vch. \u003d (Vr. - Vpr.ch.) / 2, we find:

Vtech \u003d (21.8 - 17.2) / 2 \u003d 4.62 \u003d 2.3 (km / h)

Vc \u003d Vpr tech. + Vtech \u003d 17.2 + 2.3 \u003d 19.5 (km / h)

Answer: Vc=19.5 (km/h), Vtech=2.3 (km/h).

Example 2. The steamboat passed 24 km against the current and returned back, having spent 20 minutes less on the way back than when moving against the current. Find its own speed in still water if the current speed is 3 km/h.

For X we take the own speed of the ship. Let's make a table where we will enter all the data.

Against flow With the flow

Distance 24 24

Speed ​​X-3 X+3

time 24/ (X-3) 24/ (X+3)

Knowing that the steamer spent 20 minutes less time on the return trip than on the downstream trip, we compose and solve the equation.

20 min=1/3 hour.

24 / (X-3) - 24 / (X + 3) \u003d 1/3

24*3(X+3) – (24*3(X-3)) – ((X-3)(X+3))=0

72X+216-72X+216-X2+9=0

Х=21(km/h) – own speed of the steamer.

Answer: 21 km/h.

note

The speed of the raft is considered equal to the speed of the reservoir.

According to the curriculum in mathematics, children are required to learn how to solve problems for movement in the original school. However, tasks of this type often cause difficulties for students. It is important that the child realizes what his own speed , speed flow, speed downstream and speed against the flow. Only under this condition will the student be able to easily solve problems for movement.

You will need

• Calculator, pen

Instruction

1. Own speed- this is speed boats or other vehicles in static water. Designate it - V own. The water in the river is in motion. So she has her speed, which is called speed th current (V current) Designate the speed of the boat along the river as V along the current, and speed against the current - V pr. tech.

2. Now remember the formulas needed to solve problems for movement: V pr. tech. = V own. – V tech.V tech.= V own. + V tech.

3. It turns out, based on these formulas, it is possible to make the following results. If the boat moves against the flow of the river, then V own. = V pr. tech. + V tech. If the boat moves with the flow, then V own. = V according to current – V tech.

4. We will solve several problems for moving along the river. Task 1. The speed of the boat in spite of the flow of the river is 12.1 km / h. Discover your own speed boats, knowing that speed river flow 2 km / h. Solution: 12.1 + 2 \u003d 14, 1 (km / h) - own speed boats. Task 2. The speed of the boat along the river is 16.3 km / h, speed river current 1.9 km/h. How many meters would this boat travel in 1 minute if it was in still water? Solution: 16.3 - 1.9 = 14.4 (km / h) - own speed boats. Convert km/h to m/min: 14.4 / 0.06 = 240 (m/min.). This means that in 1 minute the boat would pass 240 m. Task 3. Two boats set off at the same time opposite each other from 2 points. The 1st boat moved along the river, and the 2nd - against the current. They met three hours later. During this time, the 1st boat covered 42 km, and the 2nd - 39 km. Discover your own speed any boat, if it is known that speed river flow 2 km/h. Solution: 1) 42 / 3 = 14 (km/h) – speed movement along the river of the first boat. 2) 39 / 3 = 13 (km/h) - speed movement against the current of the river of the second boat. 3) 14 - 2 = 12 (km / h) - own speed first boat. 4) 13 + 2 = 15 (km/h) - own speed second boat.

Movement tasks seem difficult only at first glance. To discover, say, speed ship's movements contrary to currents, it is enough to imagine the situation expressed in the problem. Take your child on a little trip down the river and the student will learn to “click puzzles like nuts”.

You will need

• Calculator, pen.

Instruction

1. According to the current encyclopedia (dic.academic.ru), speed is a collation of the translational motion of a point (body), numerically equal to the ratio of the distance traveled S to the intermediate time t in uniform motion, i.e. V = S / t.

2. In order to detect the speed of a ship moving against the current, you need to know the ship's own speed and the speed of the current. Own speed is the speed of the ship in stagnant water, say, in a lake. Let's designate it - V own. The speed of the current is determined by how far the river carries the object per unit of time. Let's designate it - V tech.

3. In order to find the speed of the vessel moving against the current (V pr. tech.), It is necessary to subtract the speed of the current from the vessel's own speed. It turns out that we got the formula: V pr. tech. = V own. – V tech.

4. Let's find the speed of the ship against the flow of the river, if it is known that the own speed of the ship is 15.4 km / h, and the speed of the river is 3.2 km / h.15.4 - 3.2 \u003d 12.2 (km / h ) is the speed of the vessel moving against the current of the river.

5. In motion tasks, it is often necessary to convert km/h to m/s. In order to do this, it is necessary to remember that 1 km = 1000 m, 1 hour = 3600 s. Consequently, x km / h \u003d x * 1000 m / 3600 s \u003d x / 3.6 m / s. It turns out that in order to convert km / h to m / s, it is necessary to divide by 3.6. Let's say 72 km / h \u003d 72: 3.6 \u003d 20 m / s. In order to convert m / s to km / h, you must multiply by 3, 6. Let's say 30 m/s = 30 * 3.6 = 108 km/h.

6. Convert x km/h to m/min. To do this, recall that 1 km = 1000 m, 1 hour = 60 minutes. So x km/h = 1000 m / 60 min. = x / 0.06 m/min. Therefore, in order to convert km / h to m / min. must be divided by 0.06. Let's say 12 km/h = 200 m/min. In order to convert m/min. in km/h you need to multiply by 0.06. Let's say 250 m/min. = 15 km/h

Useful advice
Do not forget about the units in which you measure the speed.

Note!
Do not forget about the units in which you measure the speed. To convert km / h to m / s, you need to divide by 3.6. To convert m / s to km / h, you need to multiply by 3.6. To convert km / h to m/min. must be divided by 0.06. In order to translate m / min. in km/h, multiply by 0.06.

Useful advice
Drawing helps to solve the problem of movement.

According to the math curriculum, children should be able to solve motion problems as early as elementary school. However, tasks of this type often cause difficulties for students. It is important that the child understands what his own speed, speed flow, speed downstream and speed against the stream. Only under this condition, the student will be able to easily solve problems on movement.

You will need

• Calculator, pen

Instruction

Own speed- this is speed boat or other vehicle in still water. Designate it - V own.
The water in the river is in motion. So she has her speed, which is called speed th current (V current)
Designate the speed of the boat along the river - V along the current, and speed against the current - V pr. tech.

Now memorize the formulas needed to solve motion problems:
V pr. tech. = V own. - V tech.
V by current = V own. + V tech.

So, based on these formulas, we can draw the following conclusions.
If the boat moves against the current of the river, then V own. = V pr. tech. + V tech.
If the boat moves with the flow, then V own. = V according to current - V tech.

Let's solve several problems on the movement along the river.
Task 1. The speed of the boat against the current of the river is 12.1 km/h. Find your own speed boats, knowing that speed river current 2 km/h.
Solution: 12.1 + 2 = 14.1 (km/h) - own speed boats.
Task 2. The speed of the boat along the river is 16.3 km/h, speed river current 1.9 km/h. How many meters would this boat travel in 1 minute if it was in still water?
Solution: 16.3 - 1.9 \u003d 14.4 (km / h) - own speed boats. Convert km/h to m/min: 14.4 / 0.06 = 240 (m/min.). This means that in 1 minute the boat would travel 240 m.
Task 3. Two boats set off simultaneously towards each other from two points. The first boat moved along the river, and the second - against the current. They met three hours later. During this time, the first boat covered 42 km, and the second - 39 km.Find your own speed each boat, if it is known that speed river current 2 km/h.
Solution: 1) 42 / 3 = 14 (km/h) - speed movement along the river of the first boat.
2) 39 / 3 = 13 (km/h) - speed movement against the current of the river of the second boat.
3) 14 - 2 = 12 (km/h) - own speed first boat.
4) 13 + 2 = 15 (km/h) - own speed second boat.

it speed boat or other vehicle in still water. Designate it - V own.
The water in the river is in motion. So she has her speed, which speed yu (V current)
Designate the speed of the boat along the river - V along the current, and speed against the current - V pr. tech.

Let's solve several problems on the movement along the river.
Task 1. The speed of the boat against the current of the river is 12.1 km/h. Find your own speed boats, knowing that speed river current 2 km/h.
Solution: 12.1 + 2 = 14.1 (km/h) - own speed boats.
Task 2. The speed of the boat along the river is 16.3 km/h, speed river current 1.9 km/h. How far would this boat travel in 1 minute if it was in still water?
Solution: 16.3 - 1.9 \u003d 14.4 (km / h) - own speed boats. Convert km/h to m/min: 14.4 / 0.06 = 240 (m/min.). This means that in 1 minute the boat would travel 240 m.
Task 3. Two boats set off at the same time towards each other from two. The first boat moved along the river, and the second - against the current. They met for three hours. During this time, the first boat covered 42 km, and the second - 39 km.Find your own speed each boat, if it is known that speed river current 2 km/h.
Solution: 1) 42 / 3 = 14 (km/h) - speed movement along the river of the first boat.
2) 39 / 3 = 13 (km/h) - speed movement against the current of the river of the second boat.
3) 14 - 2 = 12 (km/h) - own speed first boat.
4) 13 + 2 = 15 (km/h) - own speed second boat.

note

Be aware of the units in which you measure speed.
Divide by 3.6 to convert km/h to m/s.
Multiply by 3.6 to convert m/s to km/h.
To convert km/h to m/min. must be divided by 0.06.
To convert m/min. in km/h, multiply by 0.06.

Useful advice

Drawing helps to solve the problem of movement.

Movement tasks seem difficult only at first glance. To find, for example, speed vessel movement against currents, it suffices to imagine the situation described in the problem. Take your child on a little trip down the river and the student will learn to "click puzzles like nuts".

You will need

• Calculator, pen.

Instruction

In order to find the speed of movement of any, you need the own speed of the ship and the speed of the current. Own speed is the speed of the ship in stagnant water, for example, in a lake. Let's designate it - V own. The speed of the current is determined by the distance the river carries per unit of time. Let's designate it - V tech.

To find the speed of the ship against the current (V pr. flow), you need to subtract the speed of the current from the ship's own speed. So, we got the formula: V pr. tech. = V own. - V tech.