# Conservation of Momentum

### Cut to the Chase

## Momentum

Momentum is defined as the mass multiplied by the velocity. **p**= *m***v**. Since the momentum depends on the velocity, it is also a vector quantity. The units for momentum are [kg.m.s^{-1}]

Momentum is the mass multiplied by the velocity.p=mv.

The rate of change in momentum, we explained in the section on Newton's laws of motion is another definition of the force. If we measure the momentum of a object over a small period of time, **F** = (*m***v** - *m***u**)/*δt*.

## Impulse

Taking the time to the left-hand-side of the equation, **F** *δt* = *m***v** - *m***u**. The force multiplied by the time has a special name, the impulse. The impulse is the change in the momentum. Impulse is a vector with units of [kg.m.s^{-1}] or since [kg.ms^{-2}] is equivalent to a Newton [N], the units of impulse are [N.s].

Impulse is the force multiplied by time the force acts on an object.I=Ft

Consider a ball that is thrown into the air. We can make the ball go higher in two ways, firstly, we can apply a larger force, **F** or we can make the same force act on the ball for a longer time.

If the force acting on the object is not constant over the time for the time in which the force acts, the impulse is given by the integral of the force with respect to time.

## Conservation of Momentum

The conservation is an important principle in physics that allows us to determine what happens in collisions or explosions. If a body *A* of mass m_{A} and velocity **u**_{A} collides with another body *B* of mass *m*_{B} and velocity **v**_{B} moving in the same direction. If *A* exerts a force **F** to the right on *B* for a time *t* then by Newton's third law, body *B* will exert an equal and opposite force to the left on *A* also for a time *t* but to the left. Thus the bodies receive equal but opposite impulse *Ft*. The changes of momentum must be equal and opposite. Therefore, the total momentum change is zero. In other words, the total momentum of *A* and *B* together remains constant. In mathematical terms:

*m*

_{A}

**u**

_{A}+

*m*

_{B}

**u**

_{B}=

*m*

_{A}

**v**

_{A}+

*m*

_{B}

**v**

_{B}

This is the general case. For motion in 1-dimension, we can consider the momenta in one direction only which allows us to drop the vector notation.

It is important to release that the momentum is conserved only when there are no external agent acting on the system that could add momentum to the system. They must be included in the system before the conservation of momentum can be applied.

A good example of this is a ball that is dropped from a height. If we are considering the ball alone, then the momentum is m_{b}v_{b} before the ball collides and -m_{b}v_{b} when the ball hits the Earth. The change in momentum is 2m_{b}v_{b} not zero. It appears to gain momentum from nowhere. However, we need to consider the change in momentum of the Earth. It moves with a velocity v_{E} after the collision. The conservation of momentum, m_{b}v_{b} + m_{E}.0 = m_{b}(-v)_{b}+m_{E}v_{E}. The Earth has to gain the same momentum as the ball but because the mass of the Earth is so large, the movement is unmeasurable.

## Elastic and Inelastic Collisions

Momentum is always conserved in any closed system, however kinetic energy is not always conserved and this brings us neatly to the subject of elastic and inelastic collisions.

- An elastic collision is a collision where the both the momentum and the kinetic energy are conserved.
- In an inelastic collision the momentum is conserved but the kinetic energy is not. In fact the kinetic energy will be lower after the collision because some energy has been used in creating sound, deformation and heat. The reality is that most collisions are inelastic to some extent. However, collisions between snooker balls come quite close to being elastic collisions.

## Collisions in 1-d.

In the flash animation above, the conservation of momentum is illustrated by the collision between two trolleys A and B. Initially, trolley B is stationary and trolley has a velocity *v*_{A} also the masses *m*_{A} and *m*_{B} are the same.

Applying the conservation of momentum gives, *m*_{A} *u*_{A}+ 0 = *m*_{A}*v*_{A} + *m*_{B}*v*_{B}. In the event that the two masses are the same, then mass *m*_{B} will move off at the same speed and mass *m*_{A} will stop.

To calculate what happens when the masses are different we need to use a second equation. *u*_{A} = *v*_{A}+*v*_{B}. This states that the incoming velocity is equal to the sum of the final velocities. Using this equation we obtain a general solutions for *v*_{A} and *v*_{B}.

When both masses are allowed to move, their final velocities are given by:

## Limiting Cases

### M_{A}>>M_{B}

In the limiting cases where a large mass collides with a much smaller mass, the final velocity of the small mass has a limiting value of twice the velocity of the large mass. This can be seen by looking at equation (3) since m_{A}+m_{B} ≈ m_{A} then v_{b}=2 (m_{A}/m_{A}) u_{A} = 2*u*_{A}.

### M_{A}<<M_{B}

In the case where a small mass collides with a much larger mass, the large mass hardly moves and the small mass rebounds with a velocity similar to its entry velocity. Looking at equation (4) this can be seen m_{A}-m_{B}~-m_{B} and m_{A}+m_{B}~m_{B} therefore, v_{A}~-1 u_{A}.

## Totally inelastic Collisions

So far we have considered the elastic collisions where both the kinetic energy and momentum are conserved. Most collisions are inelastic. How do we deal with those? We can measure the how much of the energy remains as kinetic energy after the collision. We take the ratio of the outgoing kinetic energy to the incoming kinetic energy. This is known as the coefficient of restitution.

*e*=kinetic energy after collision/kinetic energy before collision.

## Explosions

In an explosion, the inital momentum is zero, after the explosion the outgoing pieces move out in opposite directions so the momentum is conserved. The idea of an explosion is replicated somewhat less dramatical with two trolleys A and B with masses m_{A} and m_{B} respectively, which are initially held together. Momentum is given to the trolleys by a compressed spring. When the spring is released the momentum is given to the trolleys which move apart in opposite directions.

Applying the conservation of momentum, the momentum before the explosion is 0 since neither trolley is moving and after the explosion the momentum is m_{A}v_{A}-m_{B}v_{B}. Therefore, the change in momentum is m_{A}v_{A}-m_{B}-0 or m_{A}v_{A}=m_{B}v_{B}.

As the mass of trolley A is less than trolley B it moves away at a faster velocity.

## Experimental Confirmation of the Conservation of Momentum

The predictions of the conservation of momentum can be verified experimentally using gliders on an air track which essentially removes the effects of friction from the experiment. The gliders can be arranged to collide with one another. The glider's mass can be changed by adding additional mass. The velocity of the gliders is determined using a photogate which measures the time for the glider to pass through a beam of light. Two photogates are required to determine both of the glider's velocities.