Average speed of uneven motion formula. Uniformly accelerated motion. Calculation of path for uniform motion

Uniform movement is movement at a constant speed. That is, in other words, the body must travel the same distance in equal periods of time. For example, if a car covers a distance of 50 kilometers for every hour of its journey, then such movement will be uniform.

Generally, uniform motion is very rarely encountered in real life. Examples of uniform motion in nature include the rotation of the Earth around the Sun. Or, for example, the end of the second hand of a watch will also move evenly.

Calculation of speed during uniform motion

The speed of a body during uniform motion will be calculated using the following formula.

• Speed ​​= path / time.

If we denote the speed of movement by the letter V, the time of movement by the letter t, and the path traveled by the body by the letter S, we obtain the following formula.

• V=s/t.

The speed unit is 1 m/s. That is, a body travels a distance of one meter in a time equal to one second.

Movement with variable speed is called uneven movement. Most often, all bodies in nature move unevenly. For example, when a person walks somewhere, he moves unevenly, that is, his speed will change throughout the entire journey.

Calculation of speed during uneven movement

With uneven movement, the speed changes all the time, and in this case we talk about the average speed of movement.

The average speed of uneven movement is calculated by the formula

• Vcp=S/t.

From the formula for determining speed, we can obtain other formulas, for example, to calculate the distance traveled or the time that the body moved.

Calculation of path for uniform motion

To determine the path traveled by a body during uniform motion, it is necessary to multiply the speed of movement of the body by the time that this body moved.

• S=V*t.

That is, knowing the speed and time of movement, we can always find the path.

Now, we get a formula for calculating the time of movement, given the known speed of movement and the distance traveled.

Calculation of time during uniform motion

In order to determine the time of uniform motion, it is necessary to divide the distance traveled by the body by the speed with which this body moved.

• t=S/V.

The formulas obtained above will be valid if the body performed uniform motion.

When calculating the average speed of uneven movement, it is assumed that the movement was uniform. Based on this, to calculate the average speed of uneven movement, the distance or time of movement, the same formulas are used as for uniform movement.

Path calculation for uneven movement

We find that the path traveled by a body during uneven motion is equal to the product of the average speed and the time the body moved.

• S=Vcp*t

Calculation of time for uneven movement

The time required to travel a certain path during uneven movement is equal to the quotient of the path divided by the average speed of uneven movement.

• t=S/Vcp.

The graph of uniform motion in coordinates S(t) will be a straight line.

Uneven motion is considered to be movement with varying speed. Speed ​​can vary in direction. We can conclude that any movement NOT along a straight path is uneven. For example, the movement of a body in a circle, the movement of a body thrown into the distance, etc.

The speed can vary by numerical value. This movement will also be uneven. A special case of such motion is uniformly accelerated motion.

Sometimes there is uneven movement, which consists of alternating different types of movements, for example, first a bus accelerates (uniformly accelerated movement), then it moves uniformly for some time, and then stops.

Instantaneous speed

Uneven movement can only be characterized by speed. But the speed always changes! Therefore, we can only talk about speed at a given moment in time. When traveling by car, the speedometer shows you the instantaneous speed of movement every second. But in this case the time must be reduced not to a second, but a much shorter period of time must be considered!

average speed

What is average speed? It is wrong to think that you need to add up all the instantaneous velocities and divide by their number. This is the most common misconception about average speed! Average speed is divide the entire journey by the time taken. And it is not determined in any other way. If you consider the movement of a car, you can estimate its average speeds in the first half of the journey, in the second, and throughout the entire journey. Average speeds may be the same or may be different in these areas.

For average values, a horizontal line is drawn on top.

Average moving speed. Average ground speed

If the movement of a body is not rectilinear, then the distance traveled by the body will be greater than its displacement. In this case, the average moving speed differs from the average ground speed. Ground speed is a scalar.

The main thing to remember

1) Definition and types of uneven movement;
2) The difference between average and instantaneous speeds;
3) Rule for finding average speed

Often you need to solve a problem where the entire path is divided into equal sections, the average speeds on each section are given, you need to find the average speed along the entire route. The wrong decision will be if you add up the average speeds and divide by their number. Below is a formula that can be used to solve such problems.

Instantaneous speed can be determined using a motion graph. The instantaneous speed of a body at any point on the graph is determined by the slope of the tangent to the curve at the corresponding point. Instantaneous speed is the tangent of the angle of inclination of the tangent to the graph of the function.

Exercises

While driving a car, speedometer readings were taken every minute. Is it possible to determine the average speed of a car from these data?

It is impossible, since in the general case the value of the average speed is not equal to the arithmetic mean of the values ​​of the instantaneous speeds. But the path and time are not given.

What variable speed does the car's speedometer indicate?

Close to instantaneous. Close, since the period of time should be infinitely small, and when taking readings from the speedometer, it is impossible to judge time that way.

In what case are the instantaneous and average speeds equal? Why?

With uniform movement. Because the speed does not change.

The speed of movement of the hammer upon impact is 8 m/s. What speed is it: average or instantaneous?

With uneven motion, a body can travel both equal and different paths in equal periods of time.

To describe uneven motion, the concept is introduced average speed.

Average speed, by this definition, is a scalar quantity because the path and time are scalar quantities.

However, the average speed can also be determined through displacement according to the equation

The average speed of a path and the average speed of movement are two different quantities that can characterize the same movement.

When calculating average speed, a mistake is often made in that the concept of average speed is replaced by the concept of the arithmetic mean of the speed of the body in different areas of movement. To show the illegality of such a substitution, consider the problem and analyze its solution.

From point A train leaves for point B. For half the entire journey the train moves at a speed of 30 km/h, and for the second half of the journey at a speed of 50 km/h.

What is the average speed of the train on section AB?

The movement of the train on section AC and section CB is uniform. Looking at the text of the problem, you often immediately want to give the answer: υ av = 40 km/h.

Yes, because it seems to us that the formula used to calculate the arithmetic average is quite suitable for calculating the average speed.

Let's see: is it possible to use this formula and calculate the average speed by finding the half-sum of the given speeds.

To do this, let's consider a slightly different situation.

Let's say we're right and the average speed is really 40 km/h.

Then let's solve another problem.

As you can see, the problem texts are very similar, there is only a “very small” difference.

If in the first case we are talking about half the journey, then in the second case we are talking about half the time.

Obviously, point C in the second case is somewhat closer to point A than in the first case, and it is probably impossible to expect the same answers in the first and second problems.

If, when solving the second problem, we also give the answer that the average speed is equal to half the sum of the speeds in the first and second sections, we cannot be sure that we solved the problem correctly. What should I do?

The way out of the situation is as follows: the fact is that average speed is not determined through the arithmetic mean. There is a defining equation for average speed, according to which, to find the average speed in a certain area, the entire path traveled by the body must be divided by the entire time of movement:

We need to start solving the problem with the formula that determines the average speed, even if it seems to us that in some case we can use a simpler formula.

We will move from the question to known quantities.

We express the unknown quantity υ avg through other quantities – L 0 and Δ t 0 .

It turns out that both of these quantities are unknown, so we must express them in terms of other quantities. For example, in the first case: L 0 = 2 ∙ L, and Δ t 0 = Δ t 1 + Δ t 2.

Let us substitute these values, respectively, into the numerator and denominator of the original equation.

In the second case we do exactly the same. We don't know the whole path and all the time. We express them: and

It is obvious that the travel time on section AB in the second case and the travel time on section AB in the first case are different.

In the first case, since we do not know the times and we will try to express these quantities: and in the second case we express and:

We substitute the expressed quantities into the original equations.

Thus, in the first problem we have:

After transformation we get:

In the second case we get and after the transformation:

The answers, as predicted, are different, but in the second case we found that the average speed is indeed equal to half the sum of the speeds.

The question may arise: why can’t we immediately use this equation and give such an answer?

The point is that, having written down that the average speed in section AB in the second case is equal to half the sum of the speeds in the first and second sections, we would imagine not a solution to a problem, but a ready-made answer. The solution, as you can see, is quite long, and it begins with the defining equation. The fact that in this case we received the equation that we wanted to use initially is pure coincidence.

With uneven movement, the speed of a body can continuously change. With such movement, the speed at any subsequent point of the trajectory will differ from the speed at the previous point.

The speed of a body at a given moment of time and at a given point of the trajectory is called instantaneous speed.

The longer the time period Δt, the more the average speed differs from the instantaneous one. And, conversely, the shorter the time period, the less the average speed differs from the instantaneous speed of interest to us.

Let us define the instantaneous speed as the limit to which the average speed tends over an infinitesimal period of time:

If we are talking about the average speed of movement, then the instantaneous speed is a vector quantity:

If we are talking about the average speed of a path, then the instantaneous speed is a scalar quantity:

There are often cases when, during uneven motion, the speed of a body changes over equal periods of time by the same amount.

With uniform motion, the speed of a body can either decrease or increase.

If the speed of a body increases, then the movement is called uniformly accelerated, and if it decreases, it is called uniformly slow.

A characteristic of uniformly alternating motion is a physical quantity called acceleration.

Knowing the acceleration of the body and its initial speed, you can find the speed at any predetermined moment in time:

In projection onto the coordinate axis 0X, the equation will take the form: υ ​​x = υ 0 x + a x ∙ Δ t.

Key points:

Uneven movement is a movement with variable speed.

Instantaneous speed is a vector physical quantity equal to the limit of the ratio of the body's displacement to the period of time tending to zero.

If in arbitrary equal intervals of time a point traverses paths of different lengths, then the numerical value of its speed changes over time. This movement is called uneven. In this case, use a scalar quantity called average ground speed of uneven movement on this section of the trajectory. It is equal to the ratio of the distance traveled to the period of time during which this path was covered:

average speed in case of uneven movement - the ratio of the vector of movement of the body to the period of time during which this movement occurred.

To characterize changes in movement speed, the concept is introduced acceleration.

Medium acceleration uneven movement in the time interval from t to is called a vector quantity equal to the ratio of the change in speed to the time interval:

Instant acceleration or acceleration material point at time t, there will be a limit of average acceleration:

Movement occurring with constant acceleration is called equally variable.

Equation of uniformly alternating motion: .

The acceleration vector is usually decomposed into two components: tangential and centripetal acceleration.

Tangential acceleration shows the speed of change in the velocity modulus, and normal acceleration characterizes the speed of change in the direction of speed during curvilinear motion.

Full acceleration of a body is the geometric sum of the tangential and normal components:

;

.

Control questions:

1. Define uneven motion.

2. What is called uniformly alternating motion?

3. Define instantaneous speed.

4. What is the direction of the instantaneous velocity vector?

5. Define instantaneous acceleration. In what units is it measured?

6. How are the tangential and centripetal acceleration directed relative to the curvature of the trajectory?

7. Define angular velocity. Its units of measurement.

Complete the tasks:

1. Write dependency formulas:

a) rotation speed versus period;

b) angular velocity versus period;

c) angular and linear speed;

d) angular velocity versus frequency;

e) centripetal acceleration versus speed;

e) linear speed versus rotational speed;

g) linear speed versus period.

1. Uniform movement is rare. Generally, mechanical motion is motion with varying speed. A movement in which the speed of a body changes over time is called uneven.

For example, traffic moves unevenly. The bus, starting to move, increases its speed; When braking, its speed decreases. Bodies falling on the Earth's surface also move unevenly: their speed increases over time.

With uneven movement, the coordinate of the body can no longer be determined using the formula x = x 0 + v x t, since the speed of movement is not constant. The question arises: what value characterizes the speed of change in body position over time with uneven movement? This quantity is average speed.

Medium speed vWeduneven movement is a physical quantity equal to the displacement ratio sbodies by time t for which it was committed:

Average speed is vector quantity. To determine the average velocity module for practical purposes, this formula can be used only in the case when the body moves along a straight line in one direction. In all other cases, this formula is unsuitable.

Let's look at an example. It is necessary to calculate the time of arrival of the train at each station along the route. However, the movement is not linear. If you calculate the module of the average speed in the section between two stations using the above formula, the resulting value will differ from the value of the average speed at which the train was moving, since the module of the displacement vector is less than the distance traveled by the train. And the average speed of movement of this train from the starting point to the final point and back, in accordance with the above formula, is completely zero.

In practice, when determining the average speed, a value equal to path relation l In time t, during which this path was passed:

v Wed = .

She is often called average ground speed.

2. Knowing the average speed of a body at any part of the trajectory, it is impossible to determine its position at any time. Let's assume that the car traveled 300 km in 6 hours. The average speed of the car is 50 km/h. However, at the same time, he could stand for some time, move for some time at a speed of 70 km/h, for some time - at a speed of 20 km/h, etc.

Obviously, knowing the average speed of a car in 6 hours, we cannot determine its position after 1 hour, after 2 hours, after 3 hours, etc.

3. When moving, the body passes sequentially all points of the trajectory. At each point it is at certain times and has some speed.

Instantaneous speed is the speed of a body at a given moment in time or at a given point in the trajectory.

Let us assume that the body makes uneven linear motion. Let us determine the speed of movement of this body at the point O its trajectory (Fig. 21). Let us select a section on the trajectory AB, inside which there is a point O. Moving s 1 in this area the body has completed in time t 1 . The average speed in this section is v avg 1 = .

Let's reduce body movement. Let it be equal s 2, and the movement time is t 2. Then the average speed of the body during this time: v avg 2 = .Let us further reduce the movement, the average speed in this section is: v cf 3 = .

We will continue to reduce the time of movement of the body and, accordingly, its displacement. Eventually, the movement and time will become so small that a device, such as a speedometer in a car, will no longer record the change in speed and the movement over this short period of time can be considered uniform. The average speed in this area is the instantaneous speed of the body at the point O.

Thus,

instantaneous speed is a vector physical quantity equal to the ratio of small displacement D sto a short period of time D t, during which this movement was completed:

Self-test questions

1. What kind of movement is called uneven?

2. What is average speed?

3. What does average ground speed indicate?

4. Is it possible, knowing the trajectory of a body and its average speed over a certain period of time, to determine the position of the body at any moment in time?

5. What is instantaneous speed?

6. How do you understand the expressions “small movement” and “short period of time”?

Task 4

1. The car drove along Moscow streets 20 km in 0.5 hours, when leaving Moscow it stood for 15 minutes, and in the next 1 hour 15 minutes it drove 100 km around the Moscow region. At what average speed did the car move in each section and along the entire route?

2. What is the average speed of a train on a stretch between two stations if it traveled the first half of the distance between stations at an average speed of 50 km/h, and the second half at an average speed of 70 km/h?

3. What is the average speed of a train on a stretch between two stations if it traveled half the time at an average speed of 50 km/h, and the remaining time at an average speed of 70 km/h?