# COLLISIONS

Collisions involve a new concept -
**Momentum**.
This is *always* conserved in a collision, whether elastic or not. On the
other hand energy is only conserved in an *elastic* collision. We also use momentum to solve problems where mass or velocity changes but we cannot be sure that energy is conserved.

**Lecture Notes**: Collisions.pdf Collisions.one

## Momentum

Momentum
is usually given the letter **p** (except in the Ivanoff text book).

Momentum it is a vector in the same direction as velocity.

Two bodies with equal and opposite momentum will have zero momentum after colliding (Like clapping your hands together).

Momentum does not have a special unit of it's own like Force has Newtons (N) after Isaac Newton, or Power has Watts (W) after James Watt. So it's up for grabs. Maybe we could measure momentum in Lovetts (L) after...um... me! Just kidding.

**Momentum**

p = m * v

*Where*

p = Momentum
(kg.m/s) Use lower case **p**, uppercase **P** is for power.

m = Mass (kg)

v = Velocity (m/s)

Example of a change of mass but the same momentum. Since mass increases the velocity must reduce to maintain the same momentum.

### Conservation of Momentum

Conservation of Momentum states that the momentum before a collision
= momentum after a collision.

Or in maths:

m_{o} * v_{o}= m * v

In the case of 2 bodies, A and B with initial velocities V_{A0} and V_{B0} ...

m_{A0}V_{A0} + m_{B0}V_{B0} = m_{A}V_{A} + m_{B}V_{B}

Conservation of Momentum (Momentum part 2)

**Example Conservation of Momentum AND Energy**

A Newton's cradle demonstrates conservation of momentum. (Wikipedia). momentum is always conserved, but in this example, the collisions are elastic, * energy *is also conserved. The only way for that to happen is for the same mass to bounce off each end. (2 masses at half speed would conserve momentum, but not energy)

## Impulse

Impulse is a force for a certain amount of time. It turns out that Impulse = Change in momentum. Impulse is always applied to a body.

**Impulse = Change of
momentum**

F t = p_{1} - p_{0} = (mv_{1} - mv_{0}) = m (v_{1} - v_{0})

*Where*

p = Momentum
(kg.m/s)

m = Mass (kg)

v = Velocity (m/s)

F = Force (m/s)

t = Time (s)

The
units for impulse are Newton seconds (Ns). These units are
actually the SAME as the units for Momentum (kg.m/s), but we usually
show them differently to identify which is which.

Impulse can
be very handy because acceleration is not often constant in a
collision, which means we can't use the Linear Motion formulas to get
the acceleration. For example; When a body collides with a spring (the
most common model of collision), the force INCREASES with compression
of the spring, therefore acceleration CHANGES with time. In this case
the Linear Motion formulas cannot be used. Instead, we use Impulse to
obtain Force or Time, then continue.

In the next example, the change in momentum = m (v_{1} - v_{0}). But v_{1} = - v_{0 }2, so p = 2mv_{0} (Assuming a perfect spring)

Hence Impulse Ft = 2mv_{0}. Hence from
the spring modulus we can determine maximum force, hence
average force, which is the F in impulse. This gives us the time t.

## Coefficient of Restitution

Restitution is the rebound - the proportion of returning velocity compared to the initial velocity. If an object hits a wall at 10m/s and rebounds at 7m/s then the Coefficient of Restitution is 0.7. If there are multiple bodies then each velocity is scaled (multiplied) by the Coefficient of Restitution.

**Coefficient of Restitution**

e = V_{1 }/ V_{0}

or, when there are 2 bodies, A and B with initial velocities V_{A0} and V_{B0} ...

**e** (V_{A0} - V_{B0}) = (V_{B} - V_{A})

*Where*

**e** = Coefficient of Restitution. No units, of course - it is a ratio.

V_{1} = Velocity (m/s)

*Typical coefficients: Golf Ball = 0.78 to 0.83, Table Tennis ball = 0.94, Basketball on concrete floor 0.81-0.85*. Air reistance has a big effect though, especialy for lightweight objects like a table tennis ball.

Example 1

The
previous example now has a spring with adjustable Coefficient of
Restitution (COR). If COR =1, the carriage rebounds with the same
velocity. If COR=0 there is no rebound at all.

Example 2

A collision between 2 objects of different mass and speed. Momentum must be conserved (as always), but the speed is reduced by the coefficient of restitution.

There are two things happening here. Momentum is conserved (i.e. total p0=total p1), and the final velocities are reduced by the Coefficient of Restitution . (See Ivanoff Example 21.5)

#### Whiteboard

## Questions:

Homework
Assignment: Ivanoff new editionDo all questions; Chapter 21: The Impulse-Momentum Method

21.1 to 21.10 (page 283-284: Momentum)

21.11 to 21.18 (page 286-287: Restitution and Impulse)

Do all questions.