## Episode 220: Momentum and its conservation

Lesson for 16-19

- Activity time 120 minutes
- Level Advanced

This episode introduces the concept of momentum and its conservation.

Lesson Summary

- Demonstration and discussion: An introduction making plausible the idea of conservation of momentum (20 minutes)
- Student experiment: For them to find the law of conservation of momentum for themselves (40 minutes)
- Worked examples: Showing how to apply conservation of momentum in simple cases (20 minutes)
- Student questions: Momentum conservation (30 minutes)
- Discussion: Relating conservation of momentum to Newton II and III (10 minutes)

#### Discussion and demonstrations: An introduction making plausible the idea of conservation of momentum

Start by establishing experimentally the plausibility of the idea of conservation of momentum, by looking at some simple collisions first of all visually and then with some means of measuring velocities.

Demonstrate Newton’s cradle. Ask for an explanation in terms of forces, observing that if n balls are swung in, n balls swing out. This is a toy with limited possibilities, so move on to a better experimental system.

Demonstrate some collisions and explosions using trolleys on a flat bench or runway. (Alternatively use gliders on an air track.) Start with inelastic collisions, in which the trolleys stick together. Describe these as sticky

collisions. Point out that energy stored kinetically is not conserved. Try simple combinations such as trolleys of equal mass, or a single trolley colliding with a double one. How does velocity change? What quantity remains constant?

#### Safety

Remember that trolley runways and large air tracks require two people to manipulate and carry them safely. Some air track blowers also require two people to carry them.

Note that you are asking students to judge changes in velocity by eye. If the mass of the trolley doubles, its velocity halves, and so on.

It should become apparent that mass × velocity is constant is plausible. Name this quantity as momentum.

Emphasize that you are looking at events

and that you are comparing before

with after

. This will feed into the standard approach for solving numerical problems.

Now try explosions, in which the spring of one trolley is released to push the two apart. Try two single trolleys, then a single pushing on a double. What rule can you see? (The heavier (more massive) trolley comes away more slowly.) But what about momentum? Has it been created out of nothing? Emphasize the need to think of momentum as a vector (because velocity is a vector, mass is a scalar). Before the explosion, there is no momentum in the system; after, there are equal but opposite momenta, so the vector sum is zero.

Now you can state the principle of conservation of momentum in simple terms: in a sticky (inelastic) collision, the momentum of the moving object is shared between the colliding masses; in the Newton’s cradle case the momentum is clearly transferred, and in an explosion, there is no initial momentum, and the moving masses have equal but opposite momenta after the collision.

So total momentum before an event = total momentum after the event in all the cases so far.

#### Equipment

Note that with air-tracks it is very difficult to change the mass of the gliders by much as they then tend to sink and drag on the track. Trolleys are more flexible in this respect but friction effects are larger.

#### Student experiment: For them to find the law of conservation of momentum for themselves

The students can now investigate the idea of conservation of momentum experimentally. If you have sufficient apparatus it is very worthwhile getting the students to perform this experiment for themselves, especially if there is a quick and accurate way to measure speeds, such as light gates, or use of a motion sensor with a computer.

Episode 220-1: Observing collisions (Word, 56 KB)

Ask the students to look for a relationship in their mass and speed; if they find this difficult, suggest they look at the value of mass × speed. (Draw up a suitable table to help them with this.)

When students have made a few measurements, you may need to show how to calculate the total momentum for two masses,

i.e.
total momentum = *m*_{1}*v*_{1} + *m*_{2}*v*_{2} , not

(*m*_{1} + *m*_{2}) × (*v*_{1} + *v*_{2})
or any other combination of masses and velocities.

You could ask one group of students to demonstrate to the class the conservation of momentum in an inelastic collision, and another to demonstrate this for an explosion.

If time permits, ask students to extend the experiment to look at springy

(elastic) collisions. Is momentum still conserved?

#### Equipment

The closer the light gates are to the trolleys or gliders at the time of collision or explosion, the less friction will distort the results. Alternatively a computer and motion sensor could be used.

#### Worked examples: Showing how to apply conservation of momentum in simple cases

Show how to calculate momentum from values of mass and velocity. Emphasize that units are kg m s^{-1} (no special name in SI system).

Establish a sign convention (e.g. velocities to the right are positive; to the left they are negative).

Work through examples to show:

- calculation of velocity of moving mass after inelastic collision
- calculation of velocity of one mass after explosion, given velocity of the other

Emphasise the need to draw diagrams showing the situation before and after an event (collision or explosion) when solving numerical problems.

Emphasise the predictive power of the principle of conservation of momentum. Mention that it works in 3 dimensions, as well as in these simple 1-D situations.

(You could look at other examples of conservation they will have met, such as energy and electric charge. They should be aware of the utility of such conservation laws in calculations and also that they are established experimentally.)

Episode 220-2: Worked Examples (Word, 42 KB)

#### Student questions: Momentum conservation

Students should now be able to do more questions of the type described above. Elastic collisions are not dealt with until Episode 220.

Episode 220-3: Inelastic Collisions (Word, 21 KB)

#### Discussion: Relating conservation of momentum to Newton 2nd and 3rd Laws

(This is a rather abstract discussion, which you may wish to omit.)

The principle of conservation of momentum can be thought of as a consequence of Newton’s second and third laws. Try to prompt students to contribute to each stage in this argument.

Think about two trolleys of different masses exploding apart. From Newton’s third law it is clear that the trolleys are acted on by equal forces, in opposite directions.

Both forces must act for the same time – the time the trolleys are in contact.

These forces produce accelerations in inverse proportion to the masses, from Newton’s second law. So the bigger trolley has a smaller acceleration than the smaller one.

So the change in velocity of the bigger trolley is less than that of the smaller trolley, and in the opposite direction.

Because change in velocity ∝ 1mass, it follows that mass × velocity change for the two trolleys is equal and opposite. So total momentum is constant.