What is the projection of impulse. Body impulse: definition and properties. Relationship between force impulse and change in p¯ value

A 22-caliber bullet has a mass of only 2 g. If you throw such a bullet to someone, he can easily catch it even without gloves. If you try to catch such a bullet flying out of the muzzle at a speed of 300 m/s, then even gloves will not help.

If a toy cart is rolling towards you, you can stop it with your toe. If a truck is rolling towards you, you should move your feet out of its path.

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.

If a toy cart is rolling towards you, you can stop it with your toe. If a truck is rolling towards you, you should move your feet out of its path.

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

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

The force impulse is numerically equal to the area of the figure 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 a constant force that 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. There are no external forces of interaction.

In the real world, such a system cannot exist; there is no way to remove all external interaction. A closed system of bodies is a physical model, just as 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. The numerical value of 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, the second body with a mass of 4 kg moves in the same direction with a speed of . What are the impulses of each body? What is the momentum of the system?

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 the bodies move in opposite directions, we obtain a vector sum of multidirectional impulses. 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

Momentum in physics

Translated from Latin, “impulse” means “push.” This physical quantity is also called “quantity of motion”. It was introduced into science around the same time when Newton's laws were discovered (at the end of the 17th century).

The branch of physics that studies the movement and interaction of material bodies is mechanics. Momentum in mechanics is a vector quantity equal to the product of a body’s mass and its speed: p=mv. The directions of the momentum and velocity vectors always coincide.

In the SI system, the unit of impulse is the impulse of a body weighing 1 kg, which moves at a speed of 1 m/s. Therefore, the SI unit of impulse is 1 kg∙m/s.

In calculation problems, projections of velocity and momentum vectors onto any axis are considered and equations for these projections are used: for example, if the x axis is selected, then the projections v(x) and p(x) are considered. By definition of momentum, these quantities are related by the relation: p(x)=mv(x).

Depending on which axis is selected and where it is directed, the projection of the momentum vector onto it can be either positive or negative.

Law of conservation of momentum

The impulses of material bodies during their physical interaction can change. For example, when two balls suspended on threads collide, their impulses change mutually: one ball can start moving from a stationary state or increase its speed, and the other, on the contrary, reduce its speed or stop. However, in a closed system, i.e. when bodies interact only with each other and are not exposed to external forces, the vector sum of the impulses of these bodies remains constant during any of their interactions and movements. This is the law of conservation of momentum. Mathematically it can be derived from Newton's laws.

The law of conservation of momentum is also applicable to systems where some external forces act on bodies, but their vector sum is zero (for example, the force of gravity is balanced by the elastic force of the surface). Conventionally, such a system can also be considered closed.

In mathematical form, the law of conservation of momentum is written as follows: p1+p2+…+p(n)=p1’+p2’+…+p(n)’ (pulses p are vectors). For a two-body system, this equation looks like p1+p2=p1’+p2’, or m1v1+m2v2=m1v1’+m2v2’. For example, in the considered case with balls, the total impulse of both balls before the interaction will be equal to the total impulse after the interaction.

1. As you know, the result of a force depends on its magnitude, point of application and direction. Indeed, the greater the force acting on the body, the greater the 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.

The impulse of force is a vector physical quantity equal to the product of force and the time 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 carriage moving at the same speed collides with a light trolley, then as a result of the interaction it will move a significantly greater 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.

p = m v.

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 speed is a relative quantity.

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 is the difference between the final and initial impulses of the body, i.e. e. change in the momentum of the body.

Thus,

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

F t = D( m v).

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

4. Suppose that two balls moving on a table collide. Any interacting bodies, 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 act.

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:

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

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 of a 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 are 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?

Given:

Solution

v 01 = 0.3 m/s

v 02 = 0.2 m/s

m 1 = 16 t

m 2 = 48 t

v 1 = v 2 = v

v 02 =

1.6104kg

4.8104kg

Let us depict in the figure the direction of movement of the platforms before and after interaction (Fig. 60).

The gravity forces acting on the platforms and the support reaction forces cancel each other out. A system of two platforms can be considered closed

vx?

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.

Any problems involving moving bodies in classical mechanics require knowledge of the concept of momentum. This article discusses this concept, provides an answer to the question of where the body's momentum vector is directed, and also provides an example of solving the problem.

Quantity of movement

To find out where the momentum vector of a body is directed, you should first of all understand its physical meaning. The term was first explained by Isaac Newton, but it is important to note that the Italian scientist Galileo Galilei had already used a similar concept in his works. To characterize a moving object, he introduced a quantity called impulse, pressure, or impulse itself (impeto in Italian). The merit of Isaac Newton lies in the fact that he was able to connect this characteristic with the forces acting on the body.

So, initially and more correctly, what most understand by the impulse of a body is called the quantity of motion. Indeed, the mathematical formula for the quantity under consideration is written in the form:

Here m is the mass of the body, v¯ is its speed. As can be seen from the formula, we are not talking about any impulse, there is only the speed of the body and its mass, that is, the amount of motion.

It is important to note that this formula does not follow from mathematical proofs or expressions. Its occurrence in physics has an exclusively intuitive, everyday character. So, any person is well aware that if a fly and a truck move at the same speed, then it will be much more difficult to stop the truck, since it has much more movement than an insect.

Where the concept of body momentum vector came from is discussed below.

Impulse of force is the reason for the change in momentum

Newton was able to connect the intuitively introduced characteristic with the second law, which bears his name.

Force impulse is a known physical quantity that is equal to the product of the applied external force to a certain body and the duration of its action. Using Newton's well-known law and assuming that force does not depend on time, we can come to the expression:

F¯ * Δt = m * a¯ * Δt.

Here Δt is the time of action of force F, a is the linear acceleration imparted by force F to a body of mass m. As is known, multiplying the acceleration of a body by the period of time over which it acts gives an increase in speed. This fact allows us to rewrite the formula above in a slightly different form:

F¯ * Δt = m * Δv¯, where Δv¯= a¯ * Δt.

The right side of the equality represents the change in momentum (see the expression in the previous paragraph). Then it will turn out:

F¯ * Δt = Δp¯, where Δp¯ = m * Δv¯.

Thus, using Newton’s law and the concept of momentum, we can come to an important conclusion: the influence of an external force on an object over a period of time leads to a change in its momentum.

Now it becomes clear why the quantity of motion is usually called impulse, because its change coincides with the impulse of force (the word “force” is usually omitted).

Vector quantity p¯

Some quantities (F¯, v¯, a¯, p¯) have a bar over them. This means that we are talking about a vector characteristic. That is, the amount of motion, just like speed, force and acceleration, in addition to the absolute value (modulus), is also described by direction.

Since each vector can be decomposed into individual components, using the Cartesian rectangular coordinate system, we can write the following equalities:

1) p¯ = m * v¯;
2) p x = m * v x ; p y = m * v y ; p z = m * v z ;
3) |p¯| = √(p x 2 + p y 2 + p z 2).

Here, the 1st expression is a vector form of representation of momentum, the 2nd set of formulas allows you to calculate each of the components of momentum p¯, knowing the corresponding components of velocity (the indices x, y, z indicate the projection of the vector onto the corresponding coordinate axis). Finally, the 3rd formula allows you to calculate the length of the impulse vector (the absolute value of the magnitude) through its components.

Where is the body's momentum vector directed?

Having considered the concept of momentum p¯ and its basic properties, we can easily answer the question posed. The body's momentum vector is directed in the same way as the linear velocity vector. Indeed, it is known from mathematics that multiplying a vector a¯ by a number k leads to the formation of a new vector b¯, which has the following properties:

its length is equal to the product of the number and the modulus of the original vector, that is, |b¯| = k * |a¯|;
it is directed in the same way as the original vector if k > 0, otherwise it will be directed opposite to a¯.

In this case, the role of vector a¯ is played by the velocity v¯, the momentum p¯ is the new vector b¯, and the number k is the mass of the body m. Since the latter is always positive (m>0), then, answering the question: what is the direction of the body’s momentum vector p¯, it should be said that it is co-directed with the velocity v¯.

Momentum change vector

It is interesting to consider another similar question: where is the vector of change in the momentum of the body, that is, Δp¯, directed. To answer this, you should use the formula obtained above:

F¯ * Δt = m * Δv¯ = Δp¯.

Based on the reasoning in the previous paragraph, we can say that the direction of change in momentum Δp¯ coincides with the direction of the force vector F¯ (Δt > 0) or with the direction of the velocity change vector Δv¯ (m > 0).

It is important here not to confuse that we are talking specifically about changes in quantities. In the general case, the vectors p¯ and Δp¯ do not coincide, since they are not related to each other in any way. For example, if the force F¯ acts against the speed v¯ of the object, then p¯ and Δp¯ will be directed in opposite directions.

Where is it important to take into account the vectorial nature of momentum?

The questions discussed above: where the vector of the body’s momentum and the vector of its change are directed, are not due to simple curiosity. The fact is that the law of conservation of momentum p¯ is satisfied for each of its components. That is, in its most complete form it is written as follows:

p x = m * v x ; p y = m * v y ; p z = m * v z .

Each component of the vector p¯ retains its value in the system of interacting objects that are not affected by external forces (Δp¯ = 0).

How to use this law and vector representations of the quantity p¯ to solve problems involving the interaction (collision) of bodies?

Problem with two balls

The figure below shows two balls of different masses flying at different angles to a horizontal line. Let the masses of the balls be m 1 = 1 kg, m 2 = 0.5 kg, their speeds v 1 = 2 m/s, v 2 = 3 m/s. It is necessary to determine the direction of the impulse after the impact of the balls, assuming that the latter is absolutely inelastic.

When starting to solve the problem, you should write down the law of constancy of momentum in vector form, that is:

p 1 ¯ + p 2 ¯ = const.

Since each component of the momentum must be conserved, we need to rewrite this expression, also taking into account that after the collision the two balls will begin to move as a single object (absolutely inelastic impact):

m 1 * v 1x + m 2 * v 2x = (m 1 + m 2) * u x ;
M 1 * v 1y + m 2 * v 2y = (m 1 + m 2) * u y .

The minus sign for the projection of the momentum of the first body onto the y-axis appeared due to its direction against the selected vector of the ordinate axis (see figure).

Now you need to express the unknown components of the velocity u, and then substitute the known values into the expressions (the corresponding projections of the velocities are determined by multiplying the magnitudes of the vectors v 1 ¯ and v 2 ¯ by trigonometric functions):

u x = (m 1 * v 1x + m 2 * v 2x) / (m 1 + m 2), v 1x = v 1 * cos(45 o); v 2x = v 2 * cos(30 o);
u x = (1 * 2 * 0.7071 + 0.5 * 3 * 0.866) / (1 + 0.5) = 1.8088 m/s;
u y = (-m 1 * v 1y + m 2 * v 2y) / (m 1 + m 2), v 1y = v 1 * sin(45 o); v 2y = v 2 * sin(30 o);
u y = (-1 * 2 * 0.7071 + 0.5 * 3 * 0.5) / (1 + 0.5) = -0.4428 m/s.

These are two components of the speed of the body after impact and the “sticking together” of the balls. Since the direction of the velocity coincides with the momentum vector p¯, the question of the problem can be answered if u¯ is determined. Its angle relative to the horizontal axis will be equal to the arctangent of the ratio of the components u y and u x:

α = arctan(-0.4428 / 1.8088) = -13.756 o.

The minus sign indicates that the momentum (velocity) after the impact will be directed downward from the x-axis.