Linear Momentum and its Conservation

# Momentum

Let's imagine a soccer player who intercepts the ball and redirects it towards the goal (see Figure 1).

Figure 1: Football player kicks the ball towards the goal

He acts on the ball with force , but it lasts for a short time . This is the time the ball is in contact with the footballer's foot. The product of force and time is called the force impulse:

With a burst of force, he diverts the ball towards the goal. He changes the direction and speed of the ball. We say that he changes its momentum.

It would be more difficult for the soccer player to change the direction and speed of the ball if it were a medicine ball. It also makes a difference to a soccer player whether he kicks a medicine ball that moves slowly or quickly. These experiences help us to understand the momentum itself: the greater the mass of the body and the greater its speed, the greater its momentum:

Momentum is also a vector. It has its own magnitude (which is equal to the product of mass and velocity) and also direction. It has the same direction as the velocity vector. When adding (or subtracting), the same rules of calculation with vectors apply to momentum, as we learned in the chapter Force as a vector, or in more mathematical material, where we discuss calculation operations between vectors.

## Impulse - momentum theorem

A force impulse and a change in momentum are closely related and derived from Newton's second law. Let's take a closer look at this relationship.

A body with mass is acted upon by a force . The body begins to move with uniform acceleration :

 We note that the acceleration is given as: We multiply both sides of the equation by : We note that the expression on the right-hand side of the equation is the change in momentum :

Also through the equations, we found out what we already suspected in the introduction: the impulse of force () is the one that causes the body to change its momentum (). We illustrate this graphically in Figure 2 below:

Figure 2: An impulse changes a body's momentum

A force impulse is equal to a change in momentum:

The change in momentum is given as:

where is the momentum before the impact of the force and is the momentum after the impact of the force.

Example

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## Law of conservation of momentum

The law of conservation of momentum states that momentum is conserved if the sum of all impulses of forces acting on a body is zero. It is conserved even if the bodies collide with each other.

But let's look at the conservation law in a little more detail. Let's first discuss the following:

• types of collisions between particles;

• the difference between internal and external impulses of forces (collisions).

### Elastic and Inelastic collisions

Let's imagine bodies in a confined space. Each body has its own momentum . During motion, the bodies touch each other, we say they collide. Collisions between bodies can be:

• elastic

• inelastic

• partially elastic.

Let's take a look at them.

#### Elastic Collision

An elastic collision (also called an ideal collision) is a collision in which the entire kinetic energy is considered to be conserved during the collision.

Example

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#### Inelastic collision

An inelastic collision is a collision in which it is considered that the bodies stick together after collision. The kinetic energy of colliding bodies is not conserved.

Example

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#### Partially elastic collision

In nature, however, collisions are never completely elastic. Let's take a ball and drop it from a certain height. If the ball were perfectly elastic, it would reach the same height as the height from which we dropped it after bouncing (disregarding air resistance). But the ball reaches a lower height after each bounce until it comes to rest on the ground. Such a collision is partially elastic since the ball does bounce, but with each bounce, some kinetic energy is converted into internal energy.

For comparison, let's take a plasticine ball. If you dropped it on the ground, it wouldn't bounce at all. The collision of the plasticine ball with the ground would be inelastic.

### External and internal force impulses

Let's go back to the billiards example. In order to be able to determine what is an internal and what is an external force impulse, we must first determine what a system is. The choice of system is completely arbitrary, usually, we choose the system in such a way that it is easiest for us to calculate or observe something happening.

In the case of billiards, we choose billiard balls for the system. Now that we have determined the system, we can determine the external and internal forces:

• an external force impulse is anything that acts on the selected system from the outside. Let's list a few such external impulses of force:

• We hit one of the balls with the stick: the hit with the stick is counted as an external impulse of force on the system. The rod is not part of our chosen system and by hitting the balls, we change the momentum. If the balls before the hit are stationary, now they are moving.

• The ball bounces off the edge. The edge is not part of our system, so the action of the edge on the ball is also an external force impulse on the system. Even in the ideal case, when the collision with the edge is elastic, the direction of the momentum changes.

• The frictional force between the ball and the base is also considered as an external force impulse. The base is not part of our system, but it causes a frictional force that reduces the total momentum.

• however, an internal force impulse is everything that happens inside the system. For example, we count the collision between two balls as an internal impulse of force, since both balls are part of our chosen system.

Let's take a closer look at the internal force impulse. Let one ball collide with another:

Figure 5: impulse during elastic collision

When the first ball with mass and velocity collides with the second (see Figure 5), which is at rest, it changes its momentum with a force impulse:

Due to the mutual interaction of forces, the second ball also acts on the first with the same and oppositely directed impulse of force:

If we add both equations, we get:

We can see that the momentum of each ball has changed, but the total momentum has not changed.

External impulses of force change the total momentum of bodies. With this, they are distinguished from the internal impulses of forces that act during collisions. These change the momentum of individual bodies within the system, while the vector sum of all momentum remains unchanged.

### Law of conservation of momentum

We enclose several moving bodies in space and let them collide with each other. As a result of collisions, the directions and speeds of motion of individual bodies change. During collisions, impulses of force act on individual bodies, which change the speed and direction of motion of the body. But such impulses occur inside a closed space (within the system), so we call them internal impulses.

Let's write down the law of conservation of momentum: if no impulse of force acts on the bodies from the outside, the total momentum of all bodies in the system is conserved. The number, size, and type of internal collisions and forces are not important:

If there is no external impulse, the momentum of the system is conserved

Let's demonstrate the momentum conservation law on the example of an inelastic collision of two bodies. Since in this case, both bodies represent a system, the collision between them is internal and therefore no external force acts on the system. What are the initial momentum before collision and the final momentum after collision?

Let's write equation (1) again:

 We note that the total external force is equal to 0: We note that: We rearrange the equation:

If there is no external force acting on the system of moving bodies, the total momentum is conserved. The total momentum of all bodies after the collision is equal to the momentum before the collision:

Example

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material editor: Gabriel Amakhabi