# Episode 221: Elastic collisions

This episode extends the idea of conservation of momentum to elastic collisions, in which, because KE is conserved, useful information can also be found by calculating the changes in KE of the colliding objects.

Summary

• Demonstration and discussion: To introduce totally elastic collisions (20 minutes)
• Student experiment: To test conservation of momentum and kinetic energy in an elastic collision (20 minutes)
• Worked examples and student questions: Calculations of final velocities in elastic collisions and loss of KE in inelastic collisions (20 minutes)
• Discussion: More abstract problems and situations which commonly cause difficulties (15 minutes)
• Demonstrations: Showing some applications such as catching a ball or finding the speed of an air rifle pellet (15 minutes)

Demonstration and discussion: To introduce totally elastic collisions
This follows a similar approach to the demonstration in episode 220.

Demonstrate ‘springy’ (elastic) collisions using trolleys, one of which has its spring-load released so its spring can “soften the collisions” (Alternatively, use air-track gliders with repelling magnets attached.)

Direct a single trolley at a second, stationary trolley. The first trolley stops, the second moves off at the speed of the first. Momentum is conserved.

Now try a light trolley colliding with a heavy one, and vice versa. What pattern is seen? A light trolley bounces back from a heavier one (its momentum is negative); a heavier one moves on, but at a slower speed.

Students may accept that momentum is conserved; alternatively, with suitable light gates you should be able to measure the initial speed of the one trolley and the final speeds of both. Momentum conservation can be shown. (Or try using a computer and motion sensor.)

Now ask: How do the trolleys know at what speed they must move? There are many combinations of velocity which conserve momentum; there must be something else going on here. Introduce the idea that kinetic energy (KE) is also involved, and that, in a springy collision, there is as much KE after as before; in other words, KE is conserved.

Ask whether KE is conserved in an explosion (obviously not; it is “created” in the explosion in the change from chemical PE to KE) and in an inelastic collision (the total amount decreases, but you may need to work through a numerical example to show this).

Discuss where KE comes from in an explosion (from energy stored in a squashed spring, chemical explosive or whatever), and where it goes to in an inelastic collision (work is done in deforming material leads to heating; some sound).

Terminology: Usually, ‘elastic’ is taken to imply that KE is conserved. In some texts, this is written as ‘perfectly elastic’. ‘Inelastic’ describes a collision in which some KE is lost. Students should learn to use these terms, rather than ‘springy’ and ‘sticky’.

Student experiment: To test conservation of momentum and kinetic energy in an elastic collision
Ask students to carry out an experiment to determine whether, in a springy collision between trolleys or gliders, momentum and KE are both truly conserved.

They can use the same approach as the experiment in Episode 220, but they will have to calculate values of KE as well as momentum.

Worked examples and student questions: Calculations of final velocities in elastic collisions and loss of KE in inelastic collisions
Work through examples involving elastic collisions to show:

• conservation of KE and momentum in an elastic collision, when all values of mass and velocity are known
• calculation of final velocities in an elastic collision

Note that the second type requires solution of simultaneous equations, one of which is a quadratic; students may find this difficult, so check that it is required by the specification that you are following.

• Non-conservation of KE in an inelastic collision.

Students can now work through more examples.

Episode 221-1: Worked examples in momentum: elastic and inelastic collisions (Word, 55 KB)

Episode 221-2: Student questions (Word, 21 KB)

Discussion: More abstract problems and situations which commonly cause difficulties
Reinforce students’ ideas by discussing some of the following:

How a rocket ship works, as a controlled explosion in which reaction mass travels backwards. The rocket needs nothing to lift off except the expended fuel.

There is a video of this with a man sitting on a cart firing a fire extinguisher on the Wake Forest University website.

The same site shows another video of two people on roller-blades throwing a cushion back and forth.

Discuss situations in which the Earth is involved, as it may appear that momentum is not conserved. Where does momentum come from or go to in these situations? It helps to think about the forces involved.

• You push a car to start it moving. (Your feet push back on the Earth, so that its momentum also changes, in the opposite direction. This is equivalent to an explosion.)
• When a ball falls, it accelerates, i.e. it gains momentum. (The Earth is also accelerated minutely in the opposite direction, so momentum is conserved. The force is gravity.)
• When a ball bounces off a wall, its momentum is reversed. (Momentum is transferred to the wall + Earth by the contact force.)
• When a ball rolls to a halt, it loses momentum. (Its momentum is transferred to the Earth via friction).

These all emphasize the need to think of the closed system with which we are concerned.

Emphasise that momentum is always conserved, but KE is not. One way to think of this is that KE is just one form of energy, so it can be transformed; there is only one form of momentum, so it cannot be transformed into anything else.

Demonstrations: Showing some applications such as catching a ball or finding the speed of an air rifle pellet
Episode 221-3: Momentum demonstrations (Word, 32 KB)

The following is a brief summary of the Resourceful Physics activities given in the link in episode 221-3: Momentum demonstrations.

RP 13: Pendulum on a trolley – to show conservation of momentum. This could be filmed with a webcam and the film studied in slow motion to great effect. There is a link here between the motion of the trolley and the motion of a rowing boat, which shoots forwards when the rowers move back on their slides, rather than when they pull on their oars.

RP 18: Momentum in catching – compare to deforming barriers and elastic ones. This contrasts the momentum change on catching something and the greater change on reflecting it back.

RP 10: If you have access to an air rifle demonstration, this is a lovely way to round off the topic.

RP 24: Speed of a cricket ball can be used in place of the air rifle experiment