- State that momentum is the product of mass and velocity.
- State that the law of conservation of linear momentum can be applied to the interaction of two objects moving in one dimension, in the absence of net external forces.
- State that an elastic collision is one in which both momentum and kinetic energy are conserved.
- State that in an inelastic collision momentum is conserved but kinetic energy is not.
- Carry out calculations concerned with collisions in which the objects move in only one dimension.
- Carry out calculations concerned with explosions in one dimension.
- Apply the law of conservation of momentum to the interaction of two objects moving in one dimension to show that:

a) the changes in momentum of each object are equal in size and opposite in direction.

b) the forces acting on each object are equal in size and opposite in direction. - State that impulse = force × time.
- State that impulse = change of momentum.
- Carry out calculations involving the relationships between impulse, force, time and momentum.

Aids to Understanding

Vectorial Issues

Classifying physical quantities as vectors is extremely useful but only if we think vectorially. For example, velocity, acceleration and momentum are vectors and have both magnitude and direction. Now if these quantities change it should be clear that it can be either the magnitude or direction of the quantity that has changed. You will be tempted to think of a change as either an increase or decrease, and when working with scalars, this is fine, however you should avoid thinking in in terms of increasing and decreasing when working with vectors. One example is the motion of the Earth round the sun. The velocity of the Earth is changing by virtue of a changing direction and not due to an increasing or decreasing magnitude. This point will be clearer when discussing certain momentum contexts.

Elastic and Inelastic collisions

The elasticity of a collision depends on the ability
of the colliding objects to restore themselves to their
original shapes after the collision. Snooker balls
can clearly do this so more often than not a collision
between two of them tends to be elastic (Kinetic energy
is conserved). However if two cars crash and crumple up
then energy of the collision is used deforming
the cars and the collision is inelastic.

If two bodies
join and move off after a collision kinetic energy is
never conserved but can you prove this?
Momentum is always conserved because each colliding body
exerts an equal and opposite force on each other.

Momentum Problem

Look at the problem under the heading 5) Colliding bodies. The velocity
of trolley B was chosen to be 2.0 ms_{-1}, and was chosen
arbitralily. If we change the rubber stoppers on each car to be more or
less rubbery then this speed will change. Suppose the velocity of car B after
the collision is 1.5 ms^{-1}. Repeat the problem now.

Conservation of Momentum and the pendulum

a) Suppose an arrow of mass 200 g and travelling at
100 ms^{-1} hits and imbeds itself into
a stationary 10 kg block of wood hanging from a roof. See diagram below. To what
height will the block rise?

Answer 19.6 m.

b) Suppose a ball of mass 200 g and travelling at 100 msAnswer 24.7 m.

Impulse and Work

The following example should clear up the difference between these two concepts.
**Example:** Suppose a force of 6 N pulls a 3kg trolley for 10 s. During this time suppose it travels
20 m and its initial velocity is 2ms^{-1}

i) Calculate the final momentum of the trolley

ii) Calculate the final kinetic energy of the trolley.

iii) If the force is constant, draw a force-time graph and a force - displacement graph for the trolley and indicate the physical
quantities that the areas under these graphs mean

Answers

i) Ft = DP = mv - mumv = mu + Ft

3x(-2) + 6x10 = 54 Kgms

ii) Fd = DEk = 1/2mv

1/2mv

= 90 + 6 = 96 J.

Cumulative problem 1

Forces are fundamental concepts in physics and velocity-time generated by data loggers and computer programmes allow us to find the value of contact forces that act over very small time intervals. The concept of Impulse helps us to this end.

Example:

A 250g ball is allowed to fall from a height of 1.25 m and allowed to bounce once. The idealised velocity-time graph for
the ball's motion is shown below.

Your job is to label the axes and draw an force-time graph for the ball's motion for the time the ball is in contact with the ground.
The answer is provided below, but try to do it yourself before you look at it. The successful attempt
indicates clear understanding of much of the previous work.

Cumulative problem 2

Two bodies of equal mass fall to the ground from the same height without bouncing. The only difference between these bodies is that one of them is soft
and the other is hard.

i) Draw the shapes of the v-t graph for each body, until the upwards force on the body due to the ground (F_{BG}) equals the body's weight.

ii) Draw the shapes of the F-t graph for each ball, where t is the time the body is in contact with the ground whilst the upwards force due to the ground
is greater than its weight.

Idealised force-time graphs

A falling ball reaching ground will not experience
a constant force on contact with the ground.
The force from the ground gradually increases until the ball
is at rest, then the force decreases until
the ball eventually rises. The force-time,
acceleration-time and velocity-time graphs for the ball
are shown below. The area under f-t graph is positive so
the change in momentum and acceleration must also be positive.
Note when the force on the ball is a maximum, so is
the acceleration of the ball. At this point
the ball is at its maxiumum deformation and
its velocity is zero (ready to recoil).

The order should be oiltanker, elephant and cyclist.

(1) Find the velocity of A after the collision.

(2) Verify that DP

(3) Is the collision elastic or inelastic?

(1) Find the velocity of A after the collision.

(2) Verify that DP

(3) Verify Newton's Third law of Motion, ie show that F

You may assume the time of contact is 0.01s

- Calculate DP the change in momentum of each egg
- What impulse did the ground exert on each egg Suppose the egg with no cotton wool came to rest in 0.02 s and the egg in cotton wool came to rest on 0.5s
- Calculate, F
_{EG}the force exerted on each egg by the ground and state which egg, if any, crack.

In general though, the force on the body is not constant. For example suppose a footballer kicks a 1 kg ball. If the ball was travelling towards the player with a velocity of -5 ms

- Calculate the change in momentum of the ball
- What impulse did the foot exert on the ball
- Calculate the average force exerted on the ball (assume the ball was in contact with his foot for 0.2 s)
- Draw a simplified force-time graph for the ball