Instantaneous movement speed. Average moving speed. Average ground speed

Its coordinates change. Coordinates can change quickly or slowly. Physical quantity, which characterizes the speed of change of coordinates, is called speed.

Example

average speed-- this is a vector quantity, numerically equal to displacement per unit time, and codirectional with the displacement vector: $\left\langle v\right\rangle =\frac(\triangle r)(\triangle t)$ ; $\left\langle v\right\rangle \uparrow \uparrow \triangle r$

Figure 1. Average speed co-directional with displacement

The module of the average speed along the path is equal to: $\left\langle v\right\rangle =\frac(S)(\triangle t)$

Instantaneous speed provides precise information about movement at a specific point in time. The expression “velocity of a body at a given moment in time” is not correct from the point of view of physics. However, the concept of instantaneous speed is very convenient in mathematical calculations, and is constantly used.

Instantaneous speed (or simply speed) is the limit to which the average speed $\left\langle v\right\rangle $ tends as the time interval $\triangle t$ tends to zero:

$v=(\mathop(lim)_(\triangle t) \frac(\triangle r)(\triangle t)\ )=\frac(dr)(dt)=\dot(r)$ (1)

The vector $v$ is directed tangentially to the curvilinear trajectory, since the infinitesimal (elementary) displacement dr coincides with the infinitesimal element of the trajectory ds.

Figure 2. Instantaneous velocity vector $v$

In Cartesian coordinates, equation (1) is equivalent to three equations

$\left\( \begin(array)(c) v_x=\frac(dx)(dt)=\dot(x) \\ v_y=\frac(dy)(dt)=\dot(y) \\ v_z =\frac(dz)(dt)=\dot(z) \end(array) \right.$ (2)

The modulus of the vector $v$ in this case is equal to:

$v=\left|v\right|=\sqrt(v^2_x+v^2_y+v^2_z)=\sqrt(x^2+y^2+z^2)$ (3)

The transition from Cartesian rectangular coordinates to curvilinear ones is carried out according to the rules of differentiation complex functions. Let the radius vector r be a function of curvilinear coordinates: $r=r\left(q_1,q_2,q_3\right)\ $. Then the speed $v=\frac(dr)(dt)=\sum^3_(i=1)(\frac(\partial r)(\partial q_i)\frac(\partial q_i)(\partial t))= \sum^3_(i=1)(\frac(\partial r)(\partial q_i))\dot(q_i)$

Figure 3. Displacement and instantaneous velocity in curvilinear coordinate systems

In spherical coordinates, setting $q_1=r;\ \ q_2=\varphi ;\ \ q_3=\theta $, we obtain a representation of $v$ in the following form:

$v=v_re_r+v_(\varphi )e_(\varphi )+v_(\theta )e_(\theta )$, where $v_r=\dot(r);\ \ v_(\varphi )=r\dot( \varphi )sin\theta ;;\ \ v_(\theta )=r\dot(\theta )\ ;;$ \[\dot(r)=\frac(dr)(dt);;\ \ \dot( \varphi )=\frac(d\varphi )(dt);;\ \ \dot(\theta )=\frac(d\theta )(dt); v=r\sqrt(1+(\varphi )^2sin^2\theta +(\theta )^2)\]

Instantaneous speed is the value of the derivative of the function of displacement over time at a given moment in time, and is related to elementary displacement by the following relation: $dr=v\left(t\right)dt$

Problem 1

The law of motion of a point in a straight line: $x\left(t\right)=0.15t^2-2t+8$. Find the instantaneous speed of the point 10 seconds after the start of movement.

The instantaneous speed of a point is the first derivative of the radius vector with respect to time. Therefore, for the instantaneous speed we can write:

Answer: 10 s after the start of movement, the instantaneous speed of the point is 1 m/s.

Problem 2

The motion of a material point is given by the equation~ $x=4t-0.05t^2$. Determine the moment of time $t_(rest.)$ at which the point stops, and the average ground speed $\left\langle v\right\rangle $.

Let's find the equation for instantaneous speed: $v\left(t\right)=\dot(x)\left(t\right)=4-0.1t$

Answer: The point will stop 40 seconds after it starts moving. The average speed of its movement is 0.1 m/s.

Rolling the body down an inclined plane (Fig. 2);

Rice. 2. Rolling the body down an inclined plane ()

Free fall (Fig. 3).

All these three types of movement are not uniform, that is, their speed changes. In this lesson we will look at uneven motion.

Uniform movement – mechanical movement in which a body travels the same distance in any equal periods of time (Fig. 4).

Rice. 4. Uniform movement

Movement is called uneven, in which the body travels unequal paths in equal periods of time.

Rice. 5. Uneven movement

The main task of mechanics is to determine the position of the body at any moment in time. When the body moves unevenly, the speed of the body changes, therefore, it is necessary to learn to describe the change in the speed of the body. To do this, two concepts are introduced: average speed and instantaneous speed.

The fact of a change in the speed of a body during uneven movement does not always need to be taken into account; when considering the movement of a body over a large section of the path as a whole (the speed at each moment of time is not important to us), it is convenient to introduce the concept of average speed.

For example, a delegation of schoolchildren travels from Novosibirsk to Sochi by train. The distance between these cities is railway is approximately 3300 km. The speed of the train when it just left Novosibirsk was , does this mean that in the middle of the journey the speed was like this same, but at the entrance to Sochi [M1]? Is it possible, having only these data, to say that the travel time will be (Fig. 6). Of course not, since residents of Novosibirsk know that it takes approximately 84 hours to get to Sochi.

Rice. 6. Illustration for example

When considering the movement of a body over a large section of the path as a whole, it is more convenient to introduce the concept of average speed.

Medium speed they call the ratio of the total movement that the body has made to the time during which this movement was made (Fig. 7).

Rice. 7. Average speed

This definition is not always convenient. For example, an athlete runs 400 m - exactly one lap. The athlete’s displacement is 0 (Fig. 8), but we understand that his average speed cannot be zero.

Rice. 8. Displacement is 0

In practice, the concept of average ground speed is most often used.

Average ground speed is the ratio of the total path traveled by the body to the time during which the path was traveled (Fig. 9).

Rice. 9. Average ground speed

There is another definition of average speed.

average speed- this is the speed with which a body must move uniformly in order to cover a given distance in the same time in which it passed it, moving unevenly.

From the mathematics course we know what the arithmetic mean is. For numbers 10 and 36 it will be equal to:

In order to find out the possibility of using this formula to find the average speed, let's solve the following problem.

Task

A cyclist climbs a slope at a speed of 10 km/h, spending 0.5 hours. Then it goes down at a speed of 36 km/h in 10 minutes. Find the average speed of the cyclist (Fig. 10).

Rice. 10. Illustration for the problem

Given:; ; ;

Find:

Solution:

Since the unit of measurement for these speeds is km/h, we will find the average speed in km/h. Therefore, we will not convert these problems into SI. Let's convert to hours.

The average speed is:

The full path () consists of the path up the slope () and down the slope ():

The path to climb the slope is:

The path down the slope is:

The time it takes to travel the full path is:

Answer:.

Based on the answer to the problem, we see that it is impossible to use the arithmetic mean formula to calculate the average speed.

The concept of average speed is not always useful for solving the main problem of mechanics. Returning to the problem about the train, it cannot be said that if the average speed along the entire journey of the train is equal to , then after 5 hours it will be at a distance from Novosibirsk.

The average speed measured over an infinitesimal period of time is called instantaneous speed of the body(for example: a car’s speedometer (Fig. 11) shows instantaneous speed).

Rice. 11. Car speedometer shows instantaneous speed

There is another definition of instantaneous speed.

Instantaneous speed– the speed of movement of the body at a given moment in time, the speed of the body at a given point of the trajectory (Fig. 12).

Rice. 12. Instant speed

In order to better understand this definition, let's look at an example.

Let the car move straight along a section of highway. We have a graph of the projection of displacement versus time for a given movement (Fig. 13), let’s analyze this graph.

Rice. 13. Graph of displacement projection versus time

The graph shows that the speed of the car is not constant. Let's say you need to find the instantaneous speed of a car 30 seconds after the start of observation (at the point A). Using the definition of instantaneous speed, we find the magnitude of the average speed over the time interval from to . To do this, consider a fragment of this graph (Fig. 14).

Rice. 14. Graph of displacement projection versus time

In order to check the correctness of finding the instantaneous speed, let’s find the average speed module for the time interval from to , for this we consider a fragment of the graph (Fig. 15).

Rice. 15. Graph of displacement projection versus time

We calculate the average speed over a given period of time:

We obtained two values of the instantaneous speed of the car 30 seconds after the start of observation. More accurate will be the value where the time interval is smaller, that is. If we decrease the time interval under consideration more strongly, then the instantaneous speed of the car at the point A will be determined more accurately.

Instantaneous speed is a vector quantity. Therefore, in addition to finding it (finding its module), it is necessary to know how it is directed.

(at ) – instantaneous speed

The direction of instantaneous velocity coincides with the direction of movement of the body.

If a body moves curvilinearly, then the instantaneous speed is directed tangentially to the trajectory at a given point (Fig. 16).

Exercise 1

Can instantaneous speed () change only in direction, without changing in magnitude?

Solution

To solve this, consider the following example. The body moves along a curved path (Fig. 17). Let's mark a point on the trajectory of movement A and period B. Let us note the direction of the instantaneous velocity at these points (the instantaneous velocity is directed tangentially to the trajectory point). Let the velocities and be equal in magnitude and equal to 5 m/s.

Answer: Maybe.

Task 2

Can instantaneous speed change only in magnitude, without changing in direction?

Solution

Rice. 18. Illustration for the problem

Figure 10 shows that at the point A and at the point B instantaneous speed is in the same direction. If a body moves uniformly accelerated, then .

Answer: Maybe.

In this lesson, we began to study uneven movement, that is, movement with varying speed. The characteristics of uneven motion are average and instantaneous speeds. The concept of average speed is based on the mental replacement of uneven motion with uniform motion. Sometimes the concept of average speed (as we have seen) is very convenient, but it is not suitable for solving the main problem of mechanics. Therefore, the concept of instantaneous speed is introduced.

Bibliography

G.Ya. Myakishev, B.B. Bukhovtsev, N.N. Sotsky. Physics 10. - M.: Education, 2008.
A.P. Rymkevich. Physics. Problem book 10-11. - M.: Bustard, 2006.
O.Ya. Savchenko. Physics problems. - M.: Nauka, 1988.
A.V. Peryshkin, V.V. Krauklis. Physics course. T. 1. - M.: State. teacher ed. min. education of the RSFSR, 1957.

Internet portal “School-collection.edu.ru” ().
Internet portal “Virtulab.net” ().

Homework

Questions (1-3, 5) at the end of paragraph 9 (page 24); G.Ya. Myakishev, B.B. Bukhovtsev, N.N. Sotsky. Physics 10 (see list of recommended readings)
Is it possible, knowing the average speed over a certain period of time, to find the displacement made by a body during any part of this interval?
What is the difference between instantaneous speed during uniform linear motion and instantaneous speed during uneven motion?
While driving a car, speedometer readings were taken every minute. Is it possible to determine the average speed of a car from these data?
The cyclist rode the first third of the route at a speed of 12 km per hour, the second third at a speed of 16 km per hour, and the last third at a speed of 24 km per hour. Find the average speed of the bike over the entire journey. Give your answer in km/hour

If a material point is in motion, then its coordinates undergo changes. This process can happen quickly or slowly.

Definition 1

The quantity that characterizes the speed of change of coordinate position is called speed.

Definition 2

average speed– this is a vector quantity, numerically equal to displacement per unit time, and co-directed with the displacement vector υ = ∆ r ∆ t ; υ ∆ r.

Picture 1 . Average speed is co-directional with movement

The magnitude of the average speed along the path is equal to υ = S ∆ t.

Instantaneous speed characterizes movement at a certain point in time. The expression “body speed at a given time” is considered incorrect, but applicable in mathematical calculations.

Definition 3

Instantaneous speed is the limit to which the average speed υ tends as the time interval ∆ t tends to 0:

υ = l i m ∆ t ∆ r ∆ t = d r d t = r ˙ .

The direction of the vector υ is tangent to the curvilinear trajectory, because the infinitesimal displacement d r coincides with the infinitesimal element of the trajectory d s.

Figure 2. Instantaneous velocity vector υ

The existing expression υ = l i m ∆ t ∆ r ∆ t = d r d t = r ˙ in Cartesian coordinates is identical to the proposed equations below:

υ x = d x d t = x ˙ υ y = d y d t = y ˙ υ z = d z d t = z ˙ .

The modulus of the vector υ will take the form:

υ = υ = υ x 2 + υ y 2 + υ z 2 = x 2 + y 2 + z 2 .

To move from Cartesian rectangular coordinates to curvilinear ones, the rules for differentiating complex functions are used. If the radius vector r is a function of curvilinear coordinates r = r q 1, q 2, q 3, then the speed value will be written as:

υ = d r d t = ∑ i = 1 3 ∂ r ∂ q i ∂ q i ∂ r = ∑ i = 1 3 ∂ r ∂ q i q ˙ i .

Figure 3. Displacement and instantaneous velocity in curvilinear coordinate systems

For spherical coordinates, assume that q 1 = r; q 2 = φ; q 3 = θ, then we get υ, presented in this form:

υ = υ r e r + υ φ e φ + υ θ φ θ , where υ r = r ˙ ; υ φ = r φ ˙ sin θ ; υ θ = r θ ˙ ; r ˙ = d r d t ; φ ˙ = d φ d t ; θ ˙ = d θ d t ; υ = r 1 + φ 2 sin 2 θ + θ 2 .

Definition 4

Instant speed call the value of the derivative of the function of displacement in time at a given moment, associated with elementary displacement by the relation d r = υ (t) d t

Example 1

The law of rectilinear motion of the point x (t) = 0, 15 t 2 - 2 t + 8 is given. Determine its instantaneous speed 10 seconds after the start of movement.

Solution

The instantaneous speed is usually called the first derivative of the radius vector with respect to time. Then its entry will look like:

υ (t) = x ˙ (t) = 0 . 3 t - 2 ; υ (10) = 0 . 3 × 10 - 2 = 1 m/s.

Answer: 1 m/s.

Example 2

The motion of a material point is given by the equation x = 4 t - 0.05 t 2. Calculate the moment of time t o с t when the point stops moving, and its average ground speed υ.

Solution

Let's calculate the equation for instantaneous speed and substitute numerical expressions:

υ (t) = x ˙ (t) = 4 - 0, 1 t.

4 - 0, 1 t = 0; t o s t = 40 s; υ 0 = υ (0) = 4 ; υ = ∆ υ ∆ t = 0 - 4 40 - 0 = 0.1 m/s.

Answer: set point will stop after 40 seconds; the average speed value is 0.1 m/s.

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Speed in physics means how quickly an object moves in space. This value can be different: linear, angular, average, cosmic and even superluminal. Among everyone existing varieties Instantaneous speed is also included. What kind of quantity is this, what is its formula and what actions are necessary to calculate it - this is exactly what will be discussed in our article.

Instantaneous speed: essence and concept

Even a primary school student knows how to determine the speed of movement of an object in a straight line: divide the sufficient distance traveled by the time spent on such movement. However, it is worth remembering that the result obtained in this way allows us to judge If an object moves unevenly, then certain areas along its path, the speed of movement can vary noticeably. Therefore, sometimes such a quantity as instantaneous speed is required. It allows one to judge the speed of movement of a material point at any moment of movement.

Instantaneous speed: calculation formula

This parameter is equal to the limit (denoted by limit, abbreviated as lim) of the ratio of the displacement (coordinate difference) to the period of time during which this change occurred, provided that this period of time tends to reach zero. This definition can be written as the following formula:

v = Δs/Δt for Δt → 0 or so v = lim Δt→0 (Δs/Δt)

Note that instantaneous speed is If the movement occurs in a straight line, then it changes only in magnitude, and the direction remains constant. Otherwise, the instantaneous velocity vector is directed tangentially to the trajectory of movement at each point under consideration. What is the meaning of this indicator? Instantaneous speed allows you to find out how much movement an object will make in a unit of time if, from the moment under consideration, it moves uniformly and rectilinearly.

In this case, there are no difficulties: you just need to find the ratio of the distance to the time during which it was covered by the object. In this case, the average and instantaneous speed of the body are equal. If the movement does not occur constantly, then in this case it is necessary to find out the magnitude of the acceleration and determine the instantaneous speed at each specific moment in time. When moving vertically, the influence should be taken into account. The instantaneous speed of the car can be determined using a radar or speedometer. It should be borne in mind that the displacement in some sections of the path may take a negative value.

In order to find acceleration, you can use an accelerometer or create a motion function and use the formula v=v0+a.t. If the movement begins from a state of rest, then v0 = 0. When calculating, one must take into account the fact that when the body slows down (velocity decreases), the acceleration value will have a minus sign. If an object moves, the instantaneous speed of its movement is calculated by the formula v= g.t. In this case, the initial speed is also 0.

Rolling the body down an inclined plane (Fig. 2);

Rice. 2. Rolling the body down an inclined plane ()

Free fall (Fig. 3).

All these three types of movement are not uniform, that is, their speed changes. In this lesson we will look at uneven motion.

Uniform movement - mechanical movement in which a body travels the same distance in any equal periods of time (Fig. 4).

Rice. 4. Uniform movement

Movement is called uneven, in which the body travels unequal paths in equal periods of time.

Rice. 5. Uneven movement

For example, a delegation of schoolchildren travels from Novosibirsk to Sochi by train. The distance between these cities by rail is approximately 3,300 km. The speed of the train when it just left Novosibirsk was , does this mean that in the middle of the journey the speed was like this same, but at the entrance to Sochi [M1]? Is it possible, having only these data, to say that the travel time will be (Fig. 6). Of course not, since residents of Novosibirsk know that it takes approximately 84 hours to get to Sochi.

Rice. 6. Illustration for example

When considering the movement of a body over a large section of the path as a whole, it is more convenient to introduce the concept of average speed.

Medium speed they call the ratio of the total movement that the body has made to the time during which this movement was made (Fig. 7).

Rice. 7. Average speed

This definition is not always convenient. For example, an athlete runs 400 m - exactly one lap. The athlete’s displacement is 0 (Fig. 8), but we understand that his average speed cannot be zero.

Rice. 8. Displacement is 0

In practice, the concept of average ground speed is most often used.

Average ground speed is the ratio of the total path traveled by the body to the time during which the path was traveled (Fig. 9).

Rice. 9. Average ground speed

There is another definition of average speed.

average speed- this is the speed with which a body must move uniformly in order to cover a given distance in the same time in which it passed it, moving unevenly.

From the mathematics course we know what the arithmetic mean is. For numbers 10 and 36 it will be equal to:

In order to find out the possibility of using this formula to find the average speed, let's solve the following problem.

Task

A cyclist climbs a slope at a speed of 10 km/h, spending 0.5 hours. Then it goes down at a speed of 36 km/h in 10 minutes. Find the average speed of the cyclist (Fig. 10).

Rice. 10. Illustration for the problem

Given:; ; ;

Find:

Solution:

Since the unit of measurement for these speeds is km/h, we will find the average speed in km/h. Therefore, we will not convert these problems into SI. Let's convert to hours.

The average speed is:

The full path () consists of the path up the slope () and down the slope ():

The path to climb the slope is:

The path down the slope is:

The time it takes to travel the full path is:

Answer:.

Based on the answer to the problem, we see that it is impossible to use the arithmetic mean formula to calculate the average speed.

The average speed measured over an infinitesimal period of time is called instantaneous speed of the body(for example: a car’s speedometer (Fig. 11) shows instantaneous speed).

Rice. 11. Car speedometer shows instantaneous speed

There is another definition of instantaneous speed.

Instantaneous speed– the speed of movement of the body at a given moment in time, the speed of the body at a given point of the trajectory (Fig. 12).

Rice. 12. Instant speed

To better understand this definition, let's look at an example.

Let the car move straight along a section of highway. We have a graph of the projection of displacement versus time for a given movement (Fig. 13), let’s analyze this graph.

Rice. 13. Graph of displacement projection versus time

Rice. 14. Graph of displacement projection versus time

In order to check the correctness of finding the instantaneous speed, let’s find the average speed module for the time interval from to , for this we consider a fragment of the graph (Fig. 15).

Rice. 15. Graph of displacement projection versus time

We calculate the average speed over a given period of time:

Instantaneous speed is a vector quantity. Therefore, in addition to finding it (finding its module), it is necessary to know how it is directed.

(at ) – instantaneous speed

The direction of instantaneous velocity coincides with the direction of movement of the body.

If a body moves curvilinearly, then the instantaneous speed is directed tangentially to the trajectory at a given point (Fig. 16).

Exercise 1

Can instantaneous speed () change only in direction, without changing in magnitude?

Solution

Answer: Maybe.

Task 2

Can instantaneous speed change only in magnitude, without changing in direction?

Solution

Rice. 18. Illustration for the problem

Figure 10 shows that at the point A and at the point B instantaneous speed is in the same direction. If a body moves uniformly accelerated, then .

Answer: Maybe.

Bibliography

G.Ya. Myakishev, B.B. Bukhovtsev, N.N. Sotsky. Physics 10. - M.: Education, 2008.
A.P. Rymkevich. Physics. Problem book 10-11. - M.: Bustard, 2006.
O.Ya. Savchenko. Physics problems. - M.: Nauka, 1988.
A.V. Peryshkin, V.V. Krauklis. Physics course. T. 1. - M.: State. teacher ed. min. education of the RSFSR, 1957.

Internet portal “School-collection.edu.ru” ().
Internet portal “Virtulab.net” ().

Homework

Questions (1-3, 5) at the end of paragraph 9 (page 24); G.Ya. Myakishev, B.B. Bukhovtsev, N.N. Sotsky. Physics 10 (see list of recommended readings)
Is it possible, knowing the average speed over a certain period of time, to find the displacement made by a body during any part of this interval?
What is the difference between instantaneous speed during uniform linear motion and instantaneous speed during uneven motion?
While driving a car, speedometer readings were taken every minute. Is it possible to determine the average speed of a car from these data?
The cyclist rode the first third of the route at a speed of 12 km per hour, the second third at a speed of 16 km per hour, and the last third at a speed of 24 km per hour. Find the average speed of the bike over the entire journey. Give your answer in km/hour