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## Momentum

Lesson for 16-19

Students can develop a feel for momentum through experimentation. However, it is only by dealing with it mathematically that they can see the power of prediction which comes from the principle of conservation of momentum. A mathematical approach is also needed to distinguish between the concepts of momentum and kinetic energy.

Students’ understanding of Newton’s laws of motion will deepen once they can think of them in terms of momentum.

On the whole this topic is better approached through experiment rather than through computer simulations (which could be used later to reinforce the ideas).

## Episode 219: Preparation for momentum topic

Teaching Guidance for 16-19

- Level Advanced

Students can develop a feel for momentum through experimentation. However, it is only by dealing with it mathematically that they can see the power of prediction which comes from the principle of conservation of momentum.

## Main aims of this topic

Students will:

- understand the term momentum, and the principle of conservation of momentum
- investigate momentum changes experimentally
- solve numerical problems involving collisions and explosions
- gain experience with suitable equipment for measuring speed e.g. motion sensors, light gates

## Prior knowledge

Students should have a simple understanding of Newton’s three laws of motion both at a conceptual and a mathematical level. They should also be able to calculate energy stored kinetically.

They will also need to be familiar with measuring velocities in an experimental context.

## Where this leads

Momentum is a vital concept in mechanics, and any application of physics which involves motion or collisions. This stretches from the most obvious examples of snooker balls and traffic accidents, through pile drivers, bullet-proof vests and laser-induced fusion.

Momentum is in some ways fundamental. When considering quantum physics, students will meet the idea that light has momentum (but it doesn’t make any sense to use the classical physics definition of *m**v* for light that by definition travels at the speed of light, so Einstein’s Relativity theory is needed).

When considering sub-atomic particles, they may use the equation
KE = *p*^{ 2}2*m*. They will also learn how the momentum of a particle can be read from the curvature of its track. Combining this with the energy from a calorimeter gives the mass and hence the identity of the particle.

### Up next

### Momentum and its conservation

## 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.

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### Up next

### Elastic collisions

## Episode 221: Elastic collisions

Lesson for 16-19

- Activity time 90 minutes
- Level Advanced

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

Lesson Summary

- Demonstration and discussion: To introduce totally elastic collisions (20 minutes)
- Student experiment: To test conservation of momentum and energy stored kinetically in an elastic collision (20 minutes)
- Worked examples and student questions: Calculations of final velocities in elastic collisions, some of the initial energy is stored thermally after an inelastic collision (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)

## Discussion and demonstrations: To introduce totally elastic collisions

This follows a similar approach to the demonstration in

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 energy stored kinetically is also involved, and that, in a springy collision, there is as much energy stored kinetically after as before; in other words, energy stored kinetically is conserved.

Ask whether energy stored kinetically is conserved in an explosion (obviously not; energy was stored chemically before the explosion, and it is stored stored kinetically afterwards) and in an inelastic collision (some of the energy that was stored kinetically is stored thermally after the collision).

Discuss where energy stored kinetically comes from in an explosion (from energy stored in a squashed spring, energy stored chemically), 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 energy stored kinetically is conserved. In some texts, this is written as perfectly elastic

. Inelastic

describes a collision in which some energy stored kinetically is dissipated, or stored less usefully. Students should learn to use these terms, rather than springy

and sticky

.

## Student experiment: To test conservation of momentum and energy stored kinetically in an elastic collision

Ask students to carry out an experiment to determine whether, in a springy collision between trolleys or gliders, momentum and energy stored kinetically are both truly conserved.

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

## Worked examples and student questions: Calculations of final velocities in elastic collisions and loss of energy stored kinetically in inelastic collisions

Work through examples involving elastic collisions to show:

- conservation of energy stored kinetically 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 energy stored kinetically in an inelastic collision.

Students can now work through more examples.

Episode 221-1: Worked examples in momentum: elastic/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 and 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 and energy are always conserved, but energy stored kinetically is not. Some energy stored kinetically can be stored thermally after a collision due to physical processes (such as forces deforming objects) during the collision.

## Demonstration: 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

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### Up next

### Impulse of a force

## Episode 222: Impulse of a force

Lesson for 16-19

- Activity time 90 minutes
- Level Advanced

This episode focuses on the forces involved in changing momentum, and introduces the concept of impulse.

Lesson Summary

- Discussion: To show that impulse equals change in momentum (10 minutes)
- Demonstration: A qualitative comparison of the trade-off between force and collision time (10 minutes)
- Worked example and discussion: Using the area under a force time graph (20 minutes)
- Demonstration: Force, time and impulse (20 minutes)
- Student questions: Testing understanding, including calculations of area under a graph (30 minutes)

## Discussion: To show that impulse equals change in momentum

It is obvious that, the greater a force and the longer it acts for, the greater will be its effect. Hence, the quantity *F* × *t* is important. This is known as the impulse of the force, measured in N s.

Show that it has the same units as momentum
(N s = kg m s^{-1}).

State the impulse equation, which describes the effect of a force acting on one body:

Impulse = change in momentum

*F**t* = Δ *m**v*
or
*F**t* = *m**v* − *m**u*

(You may need to explain that Δ *m**v* means the change in momentum

, and that Δ itself is not a quantity.)

At this point, you might return to the discussion at the end of Episode 219, and extend it slightly to deduce the impulse equation.

If appropriate to the specification you are following, introduce the alternative way of writing this expression: *F* = d*m**v*d *t* and explain that this is an alternative way of writing Newton’s second law.

Or without using calculus notation

*F* = *m* × *a*

*F* = *m* (*v*_{final} − *v*_{initial})*t*

*F* = *m**v*_{final} − *m**v*_{initial}*t*

*F* = *p*_{final} − *p*_{initial}*t*

i.e. *F**t* = change in momentum

## Demonstration: A qualitative comparison of the trade-off between force and collision time

The average force on an object can be reduced by increasing the time over which it acts.

Episode 222-1: Momentum demonstrations (Word, 27 KB)

RP 23: Egg and sheet demo – egg fails to break when thrown against a suspended sheet

Relate this to real world collisions such as car crashes, or jumping off a table (important safety point: never try jumping, even small heights, with stiff legs and ankles – it is remarkably easy to break a bone). NB. No-one should climb onto a stool or table in the lab. If any experimental work is undertaken, it must be done in the gym under the supervision of a trained PE teacher.

## Worked example and discussion: Using the area under a force time graph

Use a simple example involving constant force acting for a known time.

Episode 222-2: Impulse examples (Word, 37 KB)

Forces between objects are not usually constant. Ask students to think about, say, a bat striking a ball, or a collision between cars or a car hitting a wall. What would the force-time graph look like? (The force will rise to a peak and then fall, being approximately triangular.)

The impulse of the force will be given by the area under the graph, since this area represents the sum of force ´ time in each time interval.

## Demonstration: Force, time and impulse

The average force on a football can be calculated as a pupil led demonstration.

(An alternative approach is to use sensors, but this is more demanding to set up and to interpret.)

Episode 222-3: Kicking a football (Word, 128 KB)

## Student questions: Testing understanding, including calculations of area under a graph

Look for two types of question:

Numerical questions involving impulse, including force-time graphs. These will require the student to find the area under the graph, the total momentum change and possibly the average force. This latter is a particularly difficult concept, being the equivalent force which, if applied constantly over the entire contact time would produce the same impulse. This can be related to the idea of an average speed.

Descriptive questions which take these ideas and develop them in contexts such as seat belts and airbags.

Episode 222-4: Momentum questions (Word, 77 KB)

If the students are to be stretched mathematically, look for questions which use d*m**v*d *t* directly, such as the force caused by water from a hose hitting a wall, or the lift provided by the air driven down by a helicopter’s blades.