The concept of speed is one of the main concepts in kinematics.
Many people probably know that speed is a physical quantity that shows how fast (or how slowly) a moving body moves in space. Of course, we are talking about moving in the chosen reference system. Do you know, however, that not one, but three concepts of speed are used? There is speed in this moment time, called instantaneous speed, and there are two concepts of average speed for a given period of time - the average ground speed (in English speed) and the average speed of movement (in English velocity).
We will consider a material point in the coordinate system x, y, z(Fig. a).

Position A points at time t characterize by coordinates x(t), y(t), z(t), representing the three components of the radius vector ( t). The point moves, its position in the selected coordinate system changes over time - the end of the radius vector ( t) describes a curve called the trajectory of the moving point.
The trajectory described for the time interval from t before t + Δt shown in figure b.

Through B indicates the position of the point at the moment t + Δt(it is fixed by the radius vector ( t + Δt)). Let be Δs is the length of the curvilinear trajectory under consideration, i.e. the path traveled by the point in the time from t before t + Δt.
The average ground speed of a point for a given period of time is determined by the ratio

It's obvious that v p− scalar value; it is characterized only by a numerical value.
The vector shown in figure b

is called the displacement of a material point in time from t before t + Δt.
The average speed of movement for a given period of time is determined by the ratio

It's obvious that v cf− vector quantity. vector direction v cf coincides with the direction of movement Δr.
Note that in the case of rectilinear motion, the average ground speed of the moving point coincides with the modulus of the average speed in displacement.
The movement of a point along a rectilinear or curvilinear trajectory is called uniform if, in relation (1), the value vп does not depend on Δt. If, for example, we reduce Δt 2 times, then the length of the path traveled by the point Δs will decrease by 2 times. In uniform motion, a point travels a path of equal length in equal time intervals.
 Question:
Can we assume that with a uniform motion of a point from Δt does not also depend on the vector cp of the average velocity with respect to displacement?

 Answer:
This can be considered only in the case of rectilinear motion (in this case, we recall that the modulus of the average speed for displacement is equal to the average ground speed). If the uniform motion is performed along a curvilinear trajectory, then with a change in the averaging interval Δt both the modulus and the direction of the average velocity vector along the displacement will change. With uniform curvilinear motion equal time intervals Δt will correspond to different displacement vectors Δr(and hence different vectors v cf).
True, in the case of uniform motion along a circle, equal time intervals will correspond to equal values ​​of the displacement modulus |r|(and therefore equal |v cf |). But the directions of displacements (and hence the vectors v cf) and in this case will be different for the same Δt. This is seen in the figure

Where a point uniformly moving along a circle describes equal arcs in equal intervals of time AB, BC, CD. Although the displacement vectors 1 , 2 , 3 have the same modules, but their directions are different, so there is no need to talk about the equality of these vectors.
 Note
Of the two average speeds in problems, the average ground speed is usually considered, and the average travel speed is used quite rarely. However, it deserves attention, since it allows us to introduce the concept of instantaneous speed.

This article is about how to find the average speed. The definition of this concept is given, and two important particular cases of finding the average speed are considered. A detailed analysis of tasks for finding the average speed of a body from a tutor in mathematics and physics is presented.

Determination of average speed

medium speed the movement of the body is called the ratio of the path traveled by the body to the time during which the body moved:

Let's learn how to find it on the example of the following problem:

Please note that in this case this value did not coincide with the arithmetic mean of the speeds and , which is equal to:
m/s.

Special cases of finding the average speed

1. Two identical sections of the path. Let the body move the first half of the way with the speed , and the second half of the way — with the speed . It is required to find the average speed of the body.

2. Two identical movement intervals. Let the body move at a speed for a certain period of time, and then began to move at a speed for the same period of time. It is required to find the average speed of the body.

Here we got the only case when the average speed of movement coincided with the arithmetic average speeds and on two sections of the path.

Finally, let's solve the problem from the All-Russian Olympiad for schoolchildren in physics, which took place last year, which is related to the topic of our today's lesson.

The body moved with, and the average speed of movement was 4 m/s. It is known that for the last few seconds the average velocity of the same body was 10 m/s. Determine the average speed of the body for the first s of movement.

The distance traveled by the body is: m. You can also find the path that the body has traveled for the last since its movement: m. Then for the first since its movement, the body has overcome the path in m. Therefore, the average speed on this section of the path was:
m/s.

They like to offer tasks for finding the average speed of movement at the Unified State Examination and the OGE in physics, entrance exams, and olympiads. Every student should learn how to solve these problems if he plans to continue his education at the university. A knowledgeable friend, a school teacher or a tutor in mathematics and physics can help to cope with this task. Good luck with your physics studies!

Sergey Valerievich

Instruction

Consider the function f(x) = |x|. To start this unsigned modulo, that is, the graph of the function g(x) = x. This graph is a straight line passing through the origin and the angle between this straight line and the positive direction of the x-axis is 45 degrees.

Since the modulus is a non-negative value, then the part that is below the x-axis must be mirrored relative to it. For the function g(x) = x, we get that the graph after such a mapping will become similar to V. This new graph will be a graphical interpretation of the function f(x) = |x|.

Related videos

note

The graph of the module of the function will never be in the 3rd and 4th quarters, since the module cannot take negative values.

Helpful advice

If there are several modules in the function, then they need to be expanded sequentially, and then superimposed on each other. The result will be the desired graph.

Sources:

• how to graph a function with modules

Problems on kinematics in which it is necessary to calculate speed, time or the path of uniformly and rectilinearly moving bodies, are found in the school course of algebra and physics. To solve them, find in the condition the quantities that can be equalized with each other. If the condition needs to define time at a known speed, use the following instruction.

You will need

• - pen;
• - note paper.

Instruction

The simplest case is the motion of one body with a given uniform speed Yu. The distance traveled by the body is known. Find on the way: t = S / v, hour, where S is the distance, v is the average speed body.

The second - on the oncoming movement of bodies. A car is moving from point A to point B speed u 50 km/h. At the same time, a moped with speed u 30 km/h. The distance between points A and B is 100 km. Wanted to find time through which they meet.

Designate the meeting point K. Let the distance AK, which is the car, be x km. Then the path of the motorcyclist will be 100 km. It follows from the condition of the problem that time on the road, a car and a moped are the same. Write the equation: x / v \u003d (S-x) / v ', where v, v ' are and the moped. Substituting the data, solve the equation: x = 62.5 km. Now time: t = 62.5/50 = 1.25 hours or 1 hour 15 minutes.

The third example - the same conditions are given, but the car left 20 minutes later than the moped. Determine the travel time will be the car before meeting with the moped.

Write an equation similar to the previous one. But in this case time The moped's journey will be 20 minutes than that of the car. To equalize parts, subtract one third of an hour from the right side of the expression: x/v = (S-x)/v'-1/3. Find x - 56.25. Calculate time: t = 56.25/50 = 1.125 hours or 1 hour 7 minutes 30 seconds.

The fourth example is the problem of the movement of bodies in one direction. A car and a moped move from point A at the same speed. It is known that the car left half an hour later. Through what time will he catch up with the moped?

In this case, the distance traveled by vehicles will be the same. Let be time the car will travel x hours, then time the moped will travel x+0.5 hours. You have an equation: vx = v'(x+0.5). Solve the equation by plugging in the value and find x - 0.75 hours or 45 minutes.

The fifth example - a car and a moped with the same speeds are moving in the same direction, but the moped left point B, located at a distance of 10 km from point A, half an hour earlier. Calculate through what time after the start, the car will overtake the moped.

The distance traveled by the car is 10 km more. Add this difference to the rider's path and equalize the parts of the expression: vx = v'(x+0.5)-10. Substituting the speed values ​​and solving it, you get: t = 1.25 hours or 1 hour 15 minutes.

Sources:

• what is the speed of the time machine

Instruction

Calculate the average of a body moving uniformly over a segment of the path. Such speed is the easiest to calculate, since it does not change over the entire segment movements and is equal to the mean. It can be in the form: Vrd = Vav, where Vrd - speed uniform movements, and Vav is the average speed.

Calculate Average speed equally slow (uniformly accelerated) movements in this area, for which it is necessary to add the initial and final speed. Divide by two the result obtained, which is

At school, each of us came across a problem similar to the following. If the car moved part of the way at one speed, and the next segment of the road at another, how to find the average speed?

What is this value and why is it needed? Let's try to figure this out.

Speed ​​in physics is a quantity that describes the amount of distance traveled per unit of time. That is, when they say that the speed of a pedestrian is 5 km / h, this means that he travels a distance of 5 km in 1 hour.

The formula for finding speed looks like this:
V=S/t, where S is the distance traveled, t is the time.

There is no single dimension in this formula, since it describes both extremely slow and very fast processes.

For example, an artificial satellite of the Earth overcomes about 8 km in 1 second, and the tectonic plates on which the continents are located, according to scientists, diverge by only a few millimeters per year. Therefore, the dimensions of the speed can be different - km / h, m / s, mm / s, etc.

The principle is that the distance is divided by the time required to overcome the path. Do not forget about the dimension if complex calculations are carried out.

In order not to get confused and not make a mistake in the answer, all values ​​are given in the same units of measurement. If the length of the path is indicated in kilometers, and some part of it is in centimeters, then until we get unity in dimension, we will not know the correct answer.

constant speed

Description of the formula.

The simplest case in physics is uniform motion. The speed is constant, does not change throughout the journey. There are even speed constants, summarized in tables - unchanged values. For example, sound propagates in air at a speed of 340.3 m/s.

And light is the absolute champion in this regard, it has the highest speed in our Universe - 300,000 km / s. These values ​​do not change from the starting point of the movement to the end point. They depend only on the medium in which they move (air, vacuum, water, etc.).

Uniform movement is often encountered in everyday life. This is how a conveyor works in a plant or factory, a funicular on mountain routes, an elevator (with the exception of very short periods of start and stop).

The graph of such a movement is very simple and is a straight line. 1 second - 1 m, 2 seconds - 2 m, 100 seconds - 100 m. All points are on the same straight line.

uneven speed

Unfortunately, this is ideal both in life and in physics is extremely rare. Many processes take place at an uneven speed, sometimes accelerating, sometimes slowing down.

Let's imagine the movement of an ordinary intercity bus. At the beginning of the journey, it accelerates, slows down at traffic lights, or even stops altogether. Then it goes faster outside the city, but slower on the rises, and accelerates again on the descents.

If you depict this process in the form of a graph, you get a very intricate line. It is possible to determine the speed from the graph only for a specific point, but there is no general principle.

You will need a whole set of formulas, each of which is suitable only for its section of the drawing. But there is nothing terrible. To describe the movement of the bus, the average value is used.

You can find the average speed of movement using the same formula. Indeed, we know the distance between the bus stations, measured the travel time. By dividing one by the other, find the desired value.

What is it for?

Such calculations are useful to everyone. We plan our day and travel all the time. Having a dacha outside the city, it makes sense to find out the average ground speed when traveling there.

This will make it easier to plan your holiday. By learning to find this value, we can be more punctual, stop being late.

Let's return to the example proposed at the very beginning, when the car traveled part of the way at one speed, and another part at a different one. This type of task is very often used in the school curriculum. Therefore, when your child asks you to help him solve a similar issue, it will be easy for you to do it.

Adding the lengths of the sections of the path, you get the total distance. By dividing their values ​​by the speeds indicated in the initial data, it is possible to determine the time spent on each of the sections. Adding them together, we get the time spent on the whole journey.

Tasks for average speed (hereinafter referred to as SC). We have already considered tasks for rectilinear motion. I recommend to look at the articles "" and "". Typical tasks for average speed are a group of tasks for movement, they are included in the USE in mathematics, and such a task may well be in front of you at the time of the exam itself. Problems are simple and quickly solved.

The meaning is this: imagine an object of movement, such as a car. It passes certain sections of the path at different speeds. The whole journey takes some time. So: the average speed is such a constant speed with which the car would cover a given distance in the same time. That is, the formula for the average speed is as follows:

If there were two sections of the path, then

If three, then respectively:

* In the denominator we summarize the time, and in the numerator the distances traveled for the corresponding time intervals.

The car drove the first third of the track at a speed of 90 km/h, the second third at a speed of 60 km/h, and the last third at a speed of 45 km/h. Locate the vehicle's SK throughout the journey. Give your answer in km/h.

As already mentioned, it is necessary to divide the entire path by the entire time of movement. The condition says about three sections of the path. Formula:

Denote the whole let S. Then the car drove the first third of the way:

The car drove the second third of the way:

The car drove the last third of the way:

Thus

Decide for yourself:

The car drove the first third of the track at a speed of 60 km/h, the second third at a speed of 120 km/h, and the last third at a speed of 110 km/h. Locate the vehicle's SK throughout the journey. Give your answer in km/h.

The first hour the car drove at a speed of 100 km/h, the next two hours at a speed of 90 km/h, and then for two hours at a speed of 80 km/h. Locate the vehicle's SK throughout the journey. Give your answer in km/h.

The condition says about three sections of the path. We will search for the SC by the formula:

The sections of the path are not given to us, but we can easily calculate them:

The first section of the path was 1∙100 = 100 kilometers.

The second section of the path was 2∙90 = 180 kilometers.

The third section of the path was 2∙80 = 160 kilometers.

Calculate speed:

Decide for yourself:

For the first two hours the car was traveling at a speed of 50 km/h, the next hour at a speed of 100 km/h, and then for two hours at a speed of 75 km/h. Locate the vehicle's SK throughout the journey. Give your answer in km/h.

The car drove the first 120 km at a speed of 60 km/h, the next 120 km at a speed of 80 km/h, and then 150 km at a speed of 100 km/h. Locate the vehicle's SK throughout the journey. Give your answer in km/h.

It is said about three sections of the path. Formula:

The length of the sections is given. Let's determine the time that the car spent on each section: 120/60 hours were spent on the first section, 120/80 hours on the second section, and 150/100 hours on the third. Calculate speed:

Decide for yourself:

The first 190 km the car drove at a speed of 50 km/h, the next 180 km - at a speed of 90 km/h, and then 170 km - at a speed of 100 km/h. Locate the vehicle's SK throughout the journey. Give your answer in km/h.

Half the time spent on the road, the car was traveling at a speed of 74 km / h, and the second half of the time - at a speed of 66 km / h. Locate the vehicle's SK throughout the journey. Give your answer in km/h.

*There is a problem about a traveler who crossed the sea. The guys have problems with the decision. If you do not see it, then register on the site! The registration (login) button is located in the MAIN MENU of the site. After registration, log in to the site and refresh this page.

The traveler crossed the sea on a yacht with average speed 17 km/h. He flew back on a sports plane at a speed of 323 km / h. Find the traveler's average speed for the entire journey. Give your answer in km/h.

Sincerely, Alexander.

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