Body impulse. Law of conservation of momentum

and was introduced into physics by the French scientist Rene Descartes. Descartes himself called this quantity not impulse, but “quantity of motion.” The term "impulse" appeared later. Physical quantity equal to the product of mass body to its speed, called momentum body: p=m*v. Only moving things have momentum body. Unit impulse in the International System of Units is kilogram*meter per second (1kg*m/s). For impulse a fundamental law of nature is true, called the law of conservation impulse.

Instructions

To calculate the required value, it is necessary to match the units of measurement of the two quantities included in the formula. One of these quantities that determines the momentum body- weight. Mass is a measure of inertia body. How more mass body, the more difficult it is to change the speed of this body. For example, a cabinet weighing 500 kg is more difficult to move than a cabinet weighing 100 kg. And this is obvious, the resistance of the first cabinet to the force trying to change its speed is greater than that of the second. Mass is measured in kilograms (in International system units). If the mass is not given in kilograms, then it should be converted. The following measurements of this quantity are found: tons, grams, milligrams, centners, etc. Example: 6t=6000kg, 350g=0.35kg.

Another quantity on which momentum directly depends is speed. If the body is at rest (speed is zero), then the momentum equal to zero. As the speed increases, the impulse body increases. Momentum is a vector quantity having a direction that coincides with the direction of the velocity vector body. Speed ​​is measured in meters per second (1m/s). When found impulse speed should be converted to m/s when its measurement is given in km/h. To convert to m/s, you need to multiply the numerical value of the speed by a thousand and divide by three thousand six hundred. Example: 54km/h=54*1000/3600=15m/s.

So, to determine the impulse body two quantities are multiplied: mass and speed. р=m*v. Example 1. We need to find the momentum of a running man weighing 60 kg. He runs at a speed of 6 km/h. Solution: First, the speed is converted to m/s. 6 km/h=6*1000/3600=1.7 m/s. Further, according to the formula, p = 60 kg * 1.7 m / s = 100 kg * m / s. Example 2. Find the momentum of a car at rest with a mass of 6 tons. This problem cannot be solved. Momentum of the non-moving body equal to zero.

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Let's consider a problem that demonstrates the connection between a force impulse and a change in the momentum of a body.

Example. The mass of the ball is 400 g, the speed that the ball acquired after impact is 30 m/s. The force with which the foot acted on the ball was 1500 N, and the impact time was 8 ms. Find the impulse of force and the change in momentum of the body for the ball.

Change in body momentum

Example. Estimate the average force from the floor acting on the ball during impact.

1) During a strike, two forces act on the ball: ground reaction force, gravity.

The reaction force changes during the impact time, so it is possible to find the average reaction force of the floor.

2) Change in momentum body shown in the picture

3) From Newton's second law

The main thing to remember

1) Formulas for body impulse, force impulse;
2) Direction of the impulse vector;
3) Find the change in the momentum of the body

Derivation of Newton's second law in general form

Graph F(t). Variable force

Force impulse numerically equal to area figures under the graph F(t).

If the force is not constant over time, for example it increases linearly F=kt, then the momentum of this force is equal to the area of ​​the triangle. You can replace this force with this constant force, which will change the momentum of the body by the same amount in the same period of time

Average resultant force

LAW OF CONSERVATION OF MOMENTUM

Testing online

Closed system of bodies

This is a system of bodies that interact only with each other. No external forces interactions.

IN real world There cannot be such a system; there is no way to remove all external interaction. A closed system of bodies is physical model, like a material point, is a model. This is a model of a system of bodies that supposedly interact only with each other; external forces are not taken into account, they are neglected.

Law of conservation of momentum

In a closed system of bodies vector the sum of the momenta of the bodies does not change when the bodies interact. If the momentum of one body has increased, this means that at that moment the momentum of some other body (or several bodies) has decreased by exactly the same amount.

Let's consider this example. A girl and a boy are skating. A closed system of bodies - a girl and a boy (we neglect friction and other external forces). The girl stands still, her momentum is zero, since the speed is zero (see the formula for the momentum of a body). After a boy moving at a certain speed collides with a girl, she will also begin to move. Now her body has momentum. Numerical value The girl's momentum is exactly the same as the boy's momentum decreased after the collision.

One body with a mass of 20 kg moves with a speed, a second body with a mass of 4 kg moves in the same direction with a speed of . What are the impulses of each body? Why impulse is equal systems?

Impulse of a system of bodies is the vector sum of the momenta of all bodies included in the system. In our example, this is the sum of two vectors (since two bodies are considered) that are directed in the same direction, therefore

Now let's calculate the momentum of the system of bodies from the previous example if the second body moves in the opposite direction.

Since bodies move in opposite directions, we obtain the vector sum of multidirectional pulses. Read more about vector sum.

The main thing to remember

1) What is a closed system of bodies;
2) The law of conservation of momentum and its application

1. As you know, the result of a force depends on its magnitude, point of application and direction. Indeed, than more power, acting on the body, therefore greater acceleration it acquires. The direction of the acceleration also depends on the direction of the force. So, by applying a small force to the handle, we can easily open the door, but if we apply the same force near the hinges on which the door hangs, then it may not be possible to open it.

Experiments and observations indicate that the result of a force (interaction) depends not only on the modulus of the force, but also on the time of its action. Let's do an experiment. We hang a load on a thread from the tripod, to which another thread is tied from below (Fig. 59). If you pull the lower thread sharply, it will break, and the load will remain hanging on the upper thread. If you now slowly pull the bottom thread, the top thread will break.

Force impulse is a vector physical quantity, equal to the product force for the duration of its action F t .

The SI unit of impulse of force is newton second (1 N s): [Ft] = 1 N s.

The force impulse vector coincides in direction with the force vector.

2. You also know that the result of a force depends on the mass of the body on which the force acts. Thus, the greater the mass of a body, the less acceleration it acquires under the action of the same force.

Let's look at an example. Let's imagine that there is a loaded platform on the rails. A carriage moving at some speed collides with it. As a result of the collision, the platform will acquire acceleration and move a certain distance. If a car moving at the same speed collides with a light trolley, then as a result of the interaction it will move significantly longer distance than a loaded platform.

Another example. Let's assume that a bullet approaches the target at a speed of 2 m/s. The bullet will most likely bounce off the target, leaving only a small dent in it. If the bullet flies at a speed of 100 m/s, then it will pierce the target.

Thus, the result of the interaction of bodies depends on their mass and speed of movement.

The momentum of a body is a vector physical quantity equal to the product of the mass of the body and its speed.

The SI unit of momentum of a body is kilogram-meter per second(1 kg m/s): [ p] = [m][v] = 1 kg 1m/s = 1 kg m/s.

The direction of the body's momentum coincides with the direction of its speed.

Momentum is a relative quantity; its value depends on the choice of reference system. This is understandable, since relative size is the speed.

3. Let us find out how the impulse of force and the impulse of the body are related.

According to Newton's second law:

F = ma.

Substituting the expression for acceleration into this formula a= , we get:

F= , or
Ft = mv – mv 0 .

On the left side of the equation is the impulse of force; on the right side of the equality - the difference between the final and initial body impulses, t. e. change in the momentum of the body.

Thus,

the impulse of force is equal to the change in the momentum of the body.

This is a different formulation of Newton's second law. This is exactly how Newton formulated it.

4. Let's assume that two balls moving on a table collide. Any interacting bodies, in in this case balls form system. Forces act between the bodies of the system: action force F 1 and counter force F 2. At the same time, the force of action F 1 according to Newton's third law is equal to the reaction force F 2 and is directed opposite to it: F 1 = –F 2 .

The forces with which the bodies of the system interact with each other are called internal forces.

In addition to internal forces, external forces act on the bodies of the system. Thus, the interacting balls are attracted to the Earth and are acted upon by the support reaction force. These forces are in this case external forces. During movement, the balls are subject to air resistance and friction. They are also external forces in relation to the system, which in this case consists of two balls.

External forces are forces that act on the bodies of a system from other bodies.

We will consider a system of bodies that is not affected by external forces.

A closed system is a system of bodies that interact with each other and do not interact with other bodies.

In a closed system only internal forces.

5. Let us consider the interaction of two bodies that make up a closed system. Mass of the first body m 1, its speed before interaction v 01, after interaction v 1. Mass of the second body m 2, its speed before interaction v 02 , after interaction v 2 .

The forces with which bodies interact, according to the third law: F 1 = –F 2. The time of action of the forces is the same, therefore

F 1 t = –F 2 t.

For each body we write Newton’s second law:

F 1 t = m 1 v 1 – m 1 v 01 , F 2 t = m 2 v 2 – m 2 v 02 .

Since the left sides of the equalities are equal, then their right sides are equal, i.e.

m 1 v 1 – m 1 v 01 = –(m 2 v 2 – m 2 v 02).

Transforming this equality, we get:

On the left side of the equation is the sum of the momenta of the bodies before the interaction, on the right is the sum of the momenta of the bodies after the interaction. As can be seen from this equality, the momentum of each body changed during interaction, but the sum of the impulses remained unchanged.

The geometric sum of the momenta of the bodies that make up a closed system remains constant for any interactions of the bodies of this system.

This is law of conservation of momentum.

6. A closed system of bodies is a model real system. There are no systems in nature that are not affected by external forces. However, in a number of cases, systems of interacting bodies can be considered closed. This is possible in the following cases: internal forces are much greater than external forces, interaction time is short, external forces compensate each other. In addition, the projection of external forces to any direction may be equal to zero, and then the law of conservation of momentum is satisfied for the projections of the impulses of interacting bodies to this direction.

7. Example of problem solution

Two railway platforms moving towards each other at speeds of 0.3 and 0.2 m/s. The masses of the platforms are respectively 16 and 48 tons. At what speed and in what direction will the platforms move after automatic coupling?

and apply the law of conservation of momentum to it.

m 1 v 01 + m 2 v 02 = (m 1 + m 2)v.

In projections onto the axis X can be written:

m 1 v 01x + m 2 v 02x = (m 1 + m 2)v x.

Because v 01x = v 01 ; v 02x = –v 02 ; v x = – v, That m 1 v 01 – m 2 v 02 = –(m 1 + m 2)v.

Where v = – .

v= – = 0.75 m/s.

After coupling, the platforms will move in the direction in which the platform with the larger mass moved before the interaction.

Answer: v= 0.75 m/s; directed in the direction of movement of the cart with the greater mass.

Self-test questions

1. What is the impulse of a body?

2. What is called a force impulse?

3. How are the impulse of a force and the change in the momentum of a body related?

4. What system of bodies is called closed?

5. Formulate the law of conservation of momentum.

6. What are the limits of applicability of the law of conservation of momentum?

Task 17

1. What is the momentum of a body weighing 5 kg moving at a speed of 20 m/s?

2. Determine the change in momentum of a body weighing 3 kg in 5 s under the influence of a force of 20 N.

3. Determine the momentum of a car with a mass of 1.5 tons moving at a speed of 20 m/s in a reference frame associated with: a) a car stationary relative to the Earth; b) with a car moving in the same direction at the same speed; c) with a car moving at the same speed, but in the opposite direction.

4. A boy weighing 50 kg jumped from a stationary boat weighing 100 kg located in the water near the shore. At what speed did the boat move away from the shore if the boy’s speed is directed horizontally and is equal to 1 m/s?

5. A projectile weighing 5 kg, flying horizontally, exploded into two fragments. What is the speed of the projectile if a fragment weighing 2 kg at the explosion acquired a speed of 50 m/s, and a second fragment weighing 3 kg acquired a speed of 40 m/s? The velocities of the fragments are directed horizontally.

Instructions

Find the mass of the moving body and measure its motion. After its interaction with another body, the speed of the body under study will change. In this case, subtract from the final (after interaction) initial speed and multiply the difference by body mass Δp=m∙(v2-v1). Measure the instantaneous speed with a radar and the body mass with a scale. If, after the interaction, the body begins to move in the direction opposite to that in which it moved before the interaction, then the final speed will be negative. If it is positive, it has increased, if negative, it has decreased.

Since the cause of a change in the speed of any body is force, it is also the cause of a change in momentum. To calculate the change in momentum of any body, it is enough to find the momentum of the force acting on this body at some time. Using a dynamometer, measure the force that causes a body to change speed, giving it acceleration. At the same time, use a stopwatch to measure the time that this force acts on the body. If a force causes a body to move, then consider it positive, but if it slows down its movement, consider it negative. An impulse of force equal to the change in impulse will be the product of the force and the time of its action Δp=F∙Δt.

Determining instantaneous speed with a speedometer or radar If a moving body is equipped with a speedometer (), then instantaneous speed will be continuously displayed on its scale or electronic display speed V at the moment time. When observing the body with fixed point(), send a radar signal to it, an instantaneous speed body at a given moment in time.

Video on the topic

Force is a physical quantity acting on a body, which, in particular, imparts some acceleration to it. To find pulse strength, you need to determine the change in momentum, i.e. pulse but the body itself.

Instructions

Movement material point exposure to some strength or forces that give it acceleration. Application result strength a certain amount for a certain amount is the corresponding quantity. Impulse strength the measure of its action over a certain period of time is called: Pc = Fav ∆t, where Fav – average strength, acting on the body; ∆t – time interval.

Thus, pulse strength equal to change pulse and the body: Pc = ∆Pt = m (v – v0), where v0 is the initial speed; v is the final speed of the body.

The resulting equality reflects Newton’s second law as applied to inertial system reference: the derivative of the function of a material point with respect to time is equal to the quantity constant force, acting on it: Fav ∆t = ∆Pt → Fav = dPt/dt.

Total pulse a system of several bodies can change only under the influence of external forces, and its value is directly proportional to their sum. This statement is a consequence of Newton's second and third laws. Let there be three interacting bodies, then it is true: Pс1 + Pc2 + Pc3 = ∆Pт1 + ∆Pт2 + ∆Pт3, where Pci – pulse strength, acting on the body i;Pтi – pulse bodies i.

This equality shows that if the sum of external forces is zero, then the total pulse closed system of bodies is always constant, despite the fact that the internal strength their pulse s. This principle is the law of conservation pulse A. It must be taken into account that we're talking about about vector sum.