# Momentum

Newton's Law of Motion No.1: An object at rest will remain at rest, and an object in motion will continue to move at constant velocity, unless a net force is applied.

Consequences:
• Conservation of Momentum in both elastic and inelastic collisions
• Conservation of kinetic energy in elastic collisions but not inelastic.
• Conservation of angular momentum.
Examples to help visualise the concept:
• In space, free of gravitational influence, an object will move at a constant velocity. Any change in velocity of a mass would have to conserve momentum. An astronaut who throws away his toolbox, would find himself accelerated in the opposite direction to the box. The velocity of the astronaut would be the velocity of the toolbox x (mass of toolbox)/(mass of astronaut).
• $$v_a = -v_t⋅{m_t}/{m_a}$$

Newton's First Law of Motion introduces the concept of 'inertia'. Inertia is the tendency of objects to stay as they are, whether moving or standing still. To change the motion of an object, its inertia has to be overcome by a force, which changes the energy state of the object.

Newton's First Law of Motion in action: there is insufficient friction between the coin and the card to transfer the impulse to the coin. The inertia of the coin causes it to remain in place until gravity suggests otherwise.

The inertia of a moving object can be measured by its momentum, p = mv, where p is the momentum, m is the mass and v is the velocity of the object. Newton's first law explains why an impact is greater the heavier an object, and why a skater will spin faster when she pulls in her arms and legs. Billiard balls colliding provide a good example of the transfer of momentum according to the First Law.

$$p = mv$$

where p is the momentum, m is the mass of an object moving at constant velocity v

### Impulse

Both force and momentum are vectors. The force is the first derivative (change over time t) of linear momentum:

$$F↖{→} = {Δp↖{→}}/{Δt}$$

The change in linear momentum is equal to the impulse:

$$I = Δp = F⋅Δt$$

If the mass is constant:

$F↖{→}_{net} = {Δp↖{→}}/{Δt} = {mv↖{→}_f - mv↖{→}_i}/{Δt} = m{v↖{→}_f - v↖{→}_i}/{Δt} = m{Δv↖{→}}/{Δt} = ma↖{→}$

## Elastic Collisions

Newton's First Law of Motion in action: the final two balls have momentums which sum to the momentum of the first ball.

An elastic collision is one in which the kinetic energy and momentum are conserved.

$E_k_1_{total} \text"(before collision)"$

$= E_k_2_{total} \text"(after collision)"$

Therefore, $½mv_{A_1}^2 + ½mv_{B_1}^2 = ½mv_{A_2}^2 + ½mv_{B_2}^2$

where A and B are two objects before (1) and after (2) a collision.

Since momentum p = mv, it follows that the kinetic energy is: $½mv^2 = ½pv$

Therefore, if kinetic energy is conserved, momentum is also conserved.

In large objects, some energy is always lost due to friction and internal heat. However, some collisions can be made 'more elastic' by the means of springs or more elastic materials, so an understanding of the theoretical perfect elastic collision has a role.

In the case of atoms and molecules, an ideal gas is assumed to have all collisions being elastic. In other cases, perfect elasticity is not likely to be the case.

### A coin trick

A consequence of Newton's first law is that in elastic collisions both momentum and kinetic energy is conserved. In inelastic collisions, only momentum is conserved.

Try this: find four coins of the same type (if you are like me, this will only be possible on payday). Now, place three of the coins in a straight row, so they are all touching. Slide the fourth coin quickly so that it impacts the end of the row in a straight line. What do you observe?

You should see that the first coin stops dead. None of the coins move except the one on the other end, which moves away at about the speed of the first coin.

What is happening? Why doesn't the first coin's momentum become distributed amongst all the coins?

This is a demonstration of an elastic collision. It will happen with any number of equal mass objects. The first object transfers nearly all of its kinetic energy and momentum to the next object. This energy is transferred to the next object and from that one to the next, until the last object. Since the last object has no object to stop it, it takes the energy of the previous coin and converts it to kinetic energy.

Since momentuim is p = mv, kinetic energy can be expressed as $E_k = ½mv^2 = ½pv$

Therefore, if kinetic energy is conserved (same as before), the momentum must be conserved as well!

## Inelastic Collisions

A ball bouncing experiences several energy transformations.

An inelastic collision is one in which the kinetic energy is not conserved, but momentum is.

The total momentum before the collision equals the total momentum after the collision:

$$Σp_1 = Σp_2$$ $$mv_{A_1} + mv_{B_1} = mv_{A_2} + mv_{B_2}$$

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