What's the best measure of the quantity of motion in an object?

Charlie stands on a station and the full London bound through train rushes past. Alice, sitting on the London-bound train, notices Charlie as she stares out of the window of the carriage, and also, shortly afterwards, sees Bob standing in the corridor of a local stopper-train making his way towards Birmingham. They're all objects – the stopper-train containing Bob, the London-bound train seating Alice, and Charlie – of different masses and recorded as moving at different velocities by the three observers, Alice, Bob and Charlie.

But how much motion do they have? What's the best quantity of motion – the best quantity with which to record just how much motion they have?

Is it best to measure how much motion there is, what it takes to get something moving, or what it takes to stop something that's already moving?

These kinds of questions occupied clever people for quite a while in the mid-1700s. The question is not straightforward.

What's the best measure of the quantity of motion in an object? isn't an easy question to answer. The debate between the protagonists raged long and often fiercely over the centuries that it took to evolve a powerful and useful description of motion.

Some argued for

quantity of motion = mass  ×  velocity²

and others for

quantity of motion = mass  ×  velocity.

But most were agreed that the factors that increased the quantity of motion were:

• An increase in the mass.
• An increase in the velocity.

In this episode you'll look again at the energy in the kinetic store, to find out under what circumstances that is a useful measure of the quantity of motion.

quantity of motion = 12mv²

You'll also meet a new measure of motion (momentum), which turns out to be extremely useful in interactions, and therefore will be met again in episode 04.

quantity of motion = mass × velocity

Both have their place: you'll need to learn when each is useful, and therefore in which situations you might choose one rather than the other to help you get a handle on that situation and understand it. You'll often want to know what might happen next, and so the kinds of predictions that you can make from the given situations will be a major influence in deciding which you choose.

How do you add to the quantity of motion?

Just push for a while, causing a change in the velocity: that adds to the quantity of motion.

The longer you push, the greater the change in the quantity of motion of the object. You can either add to the quantity of motion, or reduce it. Notice that there's an implicit point of view here, as your feet need to be anchored to something to push. It's on that basis alone that you can claim that you add to, or take away from, the quantity of motion. The change in motion does not depend on this particular point of view, so wherever you view the action from, the change in motion won't change – it'll be invariant.

This invariant quantity, the amount by which you change the motion, is called the impulse. The duration and intensity of your interaction with the object both contribute to the impulse.

You now know that the intensity of the interaction can be modelled by isolating the object and drawing a force arrow of appropriate direction and length (and of an appropriate kind, to keep the thinking clear based on the physical basis of the interaction modelled).

The duration, as in episode 02, is the time interval between the start and finish of the interaction.

So the impulse can be calculated as: impulse = force × time.

The impulse looks just like another accumulation, this time an accumulation of force over time. Push for a longer duration, or exert a larger force, to accumulate more change in the motion.

Other accumulations caused changes in linked quantities. For example:

Acceleration accumulating over time changed the velocity. Velocity accumulating over time changed the displacement. Power accumulating over time changed the energy. Force accumulating over distance also changed the energy.

The first two are both from this topic, the second two from the SPT: Electricity and energy topic.

So what quantity associated with the motion is changed by the action of the impulse?

Paying attention to units will reveal an insight into just how your impulse has affected the motion.

Choosing an appropriate measure of motion changed by an impulse

The required measure turns out to be something new: neither the velocity nor the energy added to the kinetic store will do. To see why not, we'd suggest looking carefully at these three measures:

Impulse is measured in newton-second (force  ×  duration). Energy is measured in joule (force  ×  distance). Velocity is measured in metre second^-1.

What could impulse change?

Velocity doesn't look possible – the units of the measures look too different, but you'd better check.

impulse: newton second velocity: metre second^-1

There ought to be a more rigorous way to compare the units – and there is. Let's try it with a more likely candidate: the energy.

The energy indeed looks like a possibility – a candidate for the measure of motion changed by the impulse. That is, the impulse will change the energy ( Δ energy = impulse). Because this is an equation, the quantities on both sides ought to be equal – that is both the numbers and the units. If the units don't agree, then it doesn't matter what the numbers are: the units provide the general condition, the numbers the particular. Here we're interested in the general.

We can compare these two quantities by reducing them all to SI base units (metre, kilogram, second, etc.). This is done by looking for relationships connecting energy and force to measures in metres, kilograms and seconds (which happen to be all the base units in this case).

Recall (or check in the SPT: Energy topic and the SPT: Forces topic, and in episode 01 of this topic):

energy = force × distance

This implies a relationship amongst the units:

joule = newton × metre

Getting, there, but we still need to connect force to more fundamental measures. You may remember, from episode 01: acceleration = forcemass.

You can rearrange this, to deduce the connections between the measures: force = mass × acceleration (expressed in units newton = kilogram × metre second^-2).

So now you have reduced newton to the base units, and a simple comparison is possible, between joule and kilogram  ×  metre second^-2  ×  metre, which can be cleaned up as kilogram  ×  metre²second² .

Impulse, expressed in units, is: newton  ×  second, which can be expanded to kilogram  ×  metre second^-2  ×  second, which can be cleaned up to represent the units of impulse as kilogram  ×  metresecond .

So it doesn't matter what the numbers are – the impulse can't be equal to the change in the energy in the kinetic store.

A powerful, general, way of thinking

This is a powerful, general and rich way of reasoning, but may not be for all students at this age.

In later study it will become even more formalised as thinking with dimensions, where it can be a powerful method of checking and even discovering the possible forms of empirical relationships. The basis is very simple: a relationship must contain an equals sign, and so the measures on both sides must be equal:

(quantity₁ = quantity₂).

Once you remember that quantity is an amalgam of unit and number, you get:

quantity₁unit₁ = quantity₂unit₂ ,

and so

number₁ = number₂

only if

unit₁ = unit₂.

So it doesn't matter what the numbers are – the impulse can't be equal to the change in the energy in the kinetic store.

It's only a variation of that old adage that equations only work if you use either pears or apples, and don't try to add pears to apples.

Introducing momentum

The units of whatever is changed by the impulse are: kilogram  ×  metresecond. You can make a subtle regrouping: kilogram  ×  metresecond, and then go back to physical quantites: Δ (physical quantity) = mass × velocity. This new quantity is given a name: it is momentum. momentum = mass × velocity (or, in symbolsp = m × v).

As velocity is a vector, momentum must be a vector as well.

When you exert an impulse (I) on an object you change its momentum.: Δ →p = →I.

You can also write this out more fully as Δ m × →v = →F ×  Δ t.

Forces are not given, but something might be

This idea that something is given to an object when you push or pull on it and affect its motion is common. This is often said by people to be giving the object force. That's not true (see the SPT: Forces topic for more details), but some have argued that the possibility of building on this intuition is a good reason to teach about momentum before teaching about force. That's a possible line of development, and worth working out in detail if you decide to go down that route. But it's not the approach developed through the SPT materials, as you'd have to make major systematic changes to teaching order to incorporate the change.

Here, however, it's probably worth bringing this to students' attention, explicitly sharing this intuition, so enabling them to link what's likely to be a persistent intuition to more formal studies of motion.

On throwing balls: changing velocity and changing momentum

Throw a tennis ball harder and it leaves your hand faster. No surprise there – we're just describing an everyday occurrence in everyday language.

Now re-describe the situation using the more precise and rarefied language of physics: Exerting a force on an object changes its motion: the object is accelerated. You generalised and quantified this pattern in episode 01 of this topic, as we introduced Newton's Second Law.

So we'd hope that you can work out what happens if you switch a tennis ball for a cricket ball, increasing the mass of what you're throwing, whilst keeping everything else the same. Remember, and reason with: acceleration = forcemass.

So you'll expect a smaller acceleration (same force, larger mass).

One thing missing from this account is the time during which you exert the force. As acceleration accumulates velocity, you'd expect that to have an effect. Again a reminder: velocity = acceleration ×  Δ time (or Δ v = a ×  Δ t).

Stop for a moment and think through the two relationships, so that you can reason out, semi-quantitatively, the effects of changing (one at a time): the mass of the ball, the force exerted and the duration for which the force is exerted.

You've been reasoning that the force exerted and the duration for which it is exerted combine together to increase the release velocity, whilst increasing the mass decreases the release velocity(m × →v = →F ×  Δ t ). These four quantities are deeply involved in the analysis and we hope you'll be reminded of the connection between impulse and momentum: Δ →p = →I.

When you exert an impulse on an object you change its momentum.

When you exert a force on an object you accelerate it: accumulate the acceleration on over an interval and you change the velocity.

We suggest it's worth formalising the connection between the force-mass-acceleration approach and the impulse-momentum approach.

A formal connection, assisted by algebra

Start with the most familiar, from episode 01: Fm = a, and then add, from episode 02:a =  Δ v Δ t. Use this to replace a in the first relationship, to get Fm =  Δ v Δ t .

A little rearranging then gives:

F Δ tm =  Δ v.

This is useful for ball throwing, but you can go further, to get the previously met:

F ×  Δ t = m ×  Δ v

That's just the connection between the impulse delivered and the change in momentum of the object.

So there is a tight connection between Newton's second law and momentum, after all. In fact, Newton originally (translated to modern notation) wrote his second law as: d p d t = F .

Useful, but not really fundamental?

So far momentum looks like just one measure of motion amongst many (velocity, energy in the kinetic store) – useful for some facets, less useful for others. With the approach we've adopted so far, there seems nothing very fundamental about the quantity – it just so happens that mass  ×  velocity is a useful measure of motion – one that you can change by the accumulated action of a force acting on the object over time. However, it's a bit more fundamental than that.

There is a very deep mathematical theorem, that connects the conservation of momentum – that is, that momentum of an isolated body must be conserved – to space being homogeneous. So if you move from one place to another, and all other things stay the same, the world will behave in the same way: the rules of the universe are invariant under translation. This is an example of a symmetry argument, which is increasingly commonly in modern physics, yet rare in elementary physics teaching.

It's one of three such fundamental principles, which you can caricature as:

• It doesn't matter when you are, the rules of the universe should be the same.
• It doesn't matter where you are, the rules of the universe should be the same.
• It doesn't matter which way you're facing, the rules of the universe should be the same.

In the early 1900s, Noether showed (much more rigorously than we can dream of doing here) that these three constraints lead to the principles that:

• The conservation of energy is a necessary consequence of invariance under translation in time.
• The conservation of momentum is a necessary consequence of invariance under translation in space.
• The conservation of angular momentum is a necessary consequence of invariance under rotation in space.

So it looks as if the momentum of an object ought to always be conserved, unless you exert a force on that object. This applies to absolutely anything that can be seen as an object – anything where we don't have to worry about the internal structure or internal interactions, but can reduce the thing to a single point. (Remember the kinds of simplifications made in the SPT: Forces topic?) Such things might include cars, elephants, galaxies, trains, a volume of air containing millions of particles and maybe even a rowing eight.

Exerting a force on the object, so modelling its interactions with other objects, will change the momentum: it's only for objects isolated from their environment, and with no resultant force acting on them, that the momentum is conserved.

Changing velocity: changing energy in the kinetic store

So, momentum looks to be a useful measure of motion. What then of the energy associated with the motion of an object – and by this, of course, we mean the energy in the kinetic store? There is certainly a greater quantity of motion if an object is shown to have more energy in its kinetic store.

Again we need to take care in choosing our point of view. If you are moving alongside an object then you will record its velocity as zero. Act on the object – by exerting a force on it – and you will change its motion. From your point of view its velocity will have changed, so you will also measure a change in the energy in the kinetic store from the first situation (first snapshot) to the second situation (second snapshot). This change will depend on the force you exert, and on the distance from which you exert it (see more in the SPT: Forces topic and the SPT: Electricity and energy topic). Both the mass and the change in velocity will affect how much energy is shifted to or from the kinetic store by the action of your force. In many cases you'll not be providing the force but, given that physical substitution, you can see this in action in many everyday situations:

• A car pulls away from the traffic lights.
• A fast commuter train pulls away from your slower moving local service.
• A fisherman's catapult propels a bait ball into the river.

Of course, you can also run this kind of thought experiment backwards. Notice something moving relative to your point of view (first snapshot) and later notice that it has stopped (second snapshot). From this you can record a reduction in the energy in the kinetic store. Such a thing happens in many everyday situations:

• A cyclist coming to a halt.
• A cricketer catching a ball.
• A safety helmet bringing the wearer's head safely to rest.

In all of these cases it is pretty clear that there is a change in the motion of the object, and that this change is reflected in the energy added to or removed from the kinetic store associated with the object as motion is changed. As both the mass and the velocity (remember, this requires you to select a particular point of view) are essentially involved in the calculation of changes in energy, and both seem intuitively to be essential in measuring the quantity of motion associated with the object, the energy in the kinetic store is a second good candidate as a measure of the motion of an object.

Power in a pathway fills a store

Filling or emptying kinetic stores necessarily involves speeding up or slowing down mass. It's an accumulation of the power in the pathway over time. The power in the pathway is just the rate at which the store is filling or emptying: how much energy you are shifting to or from the store per second. But which pathway?

The filling or emptying is most directly done by exerting a force on a mass. To fill the store up quickly, exert a large force. To fill the store up slowly, exert a smaller force. But if you remember the work about pathways, and in particular the power in a pathway, you'll remember that there are always two factors that determine the power in a pathway. (There is more on this in the SPT: Electricity and energy topic.)

The second factor is the velocity of the exerted force. If Alice is pushing, it's how fast she's moving with respect to Charlie as she pushes on the object. Charlie notices the object speeding up, and records a change in the kinetic store. Alice has a choice over how she provides the power in the mechanical working pathway (power = large force × small velocity, or power = small force × large velocity).

You can think about why it might be that the velocity is the second factor by thinking again about the units involved in the calculations, just as you did when we were considering momentum as a measure of motion.

You know that the power in the pathway must be measured in watts (that is, in joules per second). The force exerted is measured in newtons, and you may recall that the force times the distance gives you the energy in joules.

power = force × velocity

This implies a relationship amongst the units: watt = newton × metre second^-1

To see if that's correct, you need to connect force to more fundamental measures.

You may remember, from episode 01: force = mass × acceleration.

Again, you can reason with units: newton is equvalent to kilogram  ×  metre second^-2. So force  ×  velocity, can be expressed in units as kilogram  ×  metre second^-2  ×  metre second^-1. Earlier on in this episode you expressed joule as : kilogram  ×  metre second^-2  ×  metre, and you'll remember that a watt is just one joule every second. Putting these together you get: kilogram  ×  metre second^-2second, which is kilogram  ×  metre second^-2  ×  metre second^-1. This is consistent: the two sets of units agree. So velocity is the correct quantity to multiply by force to set the power in the pathway.

Choosing a force; choosing a velocity: setting the power

Power is a compensated quantity (a product of two independent contributing quantities – an increase in one contribution is compensated for by a decrease in the other to maintain a steady value of the compensated quantity). You saw that Alice had a choice:

power = large force × small velocity ,or power = small force × large velocity. But that's not all: the power in the pathway can fill or empty a store, so the power can effect an accumulation energy in the store, either positive or negative. As both force and velocity are vectors, you ought to write: VectorP = →F × →v

where ⋅ represents a special kind of multiplication (a dot product) for vectors, giving a scalar as a result – remember that power is a scalar.

Even in thinking about simple, one-dimensional situations, there are four possibilities:

• Velocity is positive, force is negative.
• Velocity is negative, force is positive.
• Velocity is negative, force is negative.
• Velocity is positive, force is positive.

In the first pair of possibilities the power is changing the energy in the store in the opposite sense to that in the second pair (reversing the flow, from filling to emptying, or from emptying to filling), which, in either case, depends on the physical situation.

Filling and emptying stores by working

Here is a particular, but common, example where energy sloshes from store to store as a result of the actions of forces on mass. And, as it's physics, we start off with a simple world: no frictional forces, so no thermal stores. Only two stores are involved: a kinetic store and a gravitational store. And you can keep it simple by focusing on snapshots – just look at the start and end points.

You can increase the complexity whilst still keeping the idea of snapshots – add friction to the imagined world, and so a thermal store. Explore this to see how the height, speed and temperature are affected by how the energy is shared out amongst the stores.

In both cases, energy is shifted to or from the stores by mechanical working – that is, by forces acting on the object that's moving up and down the hill.

• The gravity force shifts energy to or from the gravity store.
• The gravity force shifts energy to or from the kinetic store.
• The slipping forces and drag forces shift energy to the thermal store.

You might remind yourself, in passing, that the frictional forces are dissipative precisely because they act only to shift energy to the thermal store.

Rates of change

Figuring out the rate at which the energy is shifted adds another degree of complexity altogether: one that is best left to post-16 studies, even if the situation is frictionless.

However, some semi-quantitative reasoning may help you get a feel for how the quantities interact, so developing more confidence in situations where the conversations in the classroom take a tricky turn. All the evidence points towards a sound understanding resting on this style of thinking, rather than on the ability to substitute numbers into equations.

The general pattern is easy to judge as it's just mechanical working. Where the force and velocity are largest, the power in the pathway will also be largest.

For the linear hill, as you might imagine, this is likely to be near the bottom of the hill, where the velocity is largest, because the force driving the ball down the slope is constant.

For the other two hills, the situation is not so simple, because the force driving the object down the hill also varies – the steeper the slope of the hill, the greater the driving force. For the convex hill, the driving force and velocity will both be largest near the bottom, and so here the power in the pathway will definitely be the largest. The concave hill will have the largest driving force at the top of the hill, but the largest velocity at the bottom. You'd need more precise data about the shape of the hill to say where the power is greatest.

You can begin to reason about situations which aren't frictionless by making connections between the power in the pathway that fills the thermal store and the velocity. Drag forces increase with velocity: sliding forces may do so as well.

Measures of motion and how the force acting can change them

Alice, Bob or Charlie can record the velocity of an object. Combining these measures with the mass of the object (invariant – so the same for all three) results in two measures of motion for any object.

Energy in the kinetic store (a scalar quantity):

energy in the kinetic store = 12mv²

Momentum (a vector quantity): →p = m × →v.

Evidently these measures depend on the point of view chosen for the recording of the velocity, so won't be invariant. Alice, Bob and Charlie will calculate different values for the quantities unless they happen to all be moving together, so having the same velocity.

All kinds of compound things can be taken as objects, so long as we don't have to consider the internal structure. That is, any internal interactions need not be considered for the purpose we have at hand, and so the things can be reduced to point masses.

You can augment either quantity by acting on the object with a force, and both are compensated quantities (one case for each is given here). energy = large force × small distance momentum = large force × short duration or: energy = small force × long distance momentum = small force × long duration