Bringing together two sets of constraints

Focusing on the learners:

Distinguishing–eliciting–connecting. How will you:

• re-activate children's ideas about forces and energy
• connect to the measures of motion to everyday concerns

Teacher Tip: These are all related to findings about children's ideas from research. The teaching activities will provide some suggestions. So will colleagues, near and far.

Focusing on the physics:

Representing–noticing–recording. How will you:

• use diagrams, words and actions that separate force, momentum and energy
• keep scalars and vectors separate
• exploit common patterns

Teacher Tip: Connecting what is experienced with what is written and drawn is essential to making sense of the connections between the theoretical world of physics and the lived-in world of the children. Don't forget to exemplify this action.

Force and energy

Wrong Track: That bus had lots of force: it had lots of energy before it hit the pillar.

Wrong Track: Kinetic energy and momentum must be the same: both get bigger as you go faster.

Right Lines: There are two useful measures of the quantity of motion of an object. Neither of these is the force. These measures are the energy in the kinetic store and the momentum.

Distinguishing between force, energy and momentum

Thinking about the learning

This challenge arises from an imprecise use of specialist terms, and perhaps from too much reliance on language alone. There are quantities that are reduced when an object is brought to rest, but none of these is the force, as a moving object does not have force.

Momentum and energy are both reduced when the object is brought to rest. The motion will stop when the energy in the kinetic store is shifted elsewhere – for example, once the brakes have been applied (a thermal store is filled as the brakes and surrounding air are warmed). A free-wheeling vehicle will eventually stop once the frictional forces have shifted all the energy from the kinetic store. In describing such behaviour as running out of force, we can see pupils with almost the right idea but mixing the terms force and energy.

Thinking about the teaching

Elsewhere we've suggested ways of introducing forces that we think will help students identify them accurately (in the SPT: Forces topic). The key point is that they are a way of describing the interactions of the object with the environment. Both momentum and the energy in the kinetic store are properly attributes of the object (at least from a particular point of view), and so could increase or decrease as the object's motion changes. Of course, one is a scalar and one a vector – a distinction probably best made by drawing.

There are fundamental differences between momentum and energy, but we'd suggest that it's far more profitable to focus on seeing how the ideas are used, as it's this experience-based skill that will bring students a good feel for the ideas.

Power in pathways fill stores

The power in a pathway can fill or empty a store, so the power can effect a accumulation of energy in the store (the accumulation can be either positive or negative). As both force and velocity are vectors, you really ought to write: VectorP = →F × →v .

Here, ⋅ 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.

With 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). In either case this depends on the physical situation: we'd recommend thinking about this rather than trying to memorise sign conventions and linking those to filling or emptying actions.

Shown by an arrow

Wrong Track: Momentum is a simple measure of the motion, like the energy in the kinetic store, and so it's just a bigger or smaller number, except that the number can sometimes be negative.

Right Lines: →p has to be shown with an arrow. It's a quantity with a direction, just like →v. Just like in mathematics, you need ordered sets of numbers to show these quantities, as they're vectors (in our world, we'd need three such numbers to cover the three dimensions up-down, left-right, and in-out).

Thinking about the learning

Momenta met in traditional problems often feel just like numbers on a number-line, because only one dimension is considered in order to simplify the problem. So it's perhaps not surprising that the vector nature of momentum is hard to get hold of from the traditional examples, where momentum is represented as either a positive or a negative number.

Thinking about the teaching

The SPT materials have been using arrows to show quantities with direction throughout. Regular use of these kinds of depictions can emphasise that the relative magnitudes don't capture all that there is to say about the quantity: directions are also important. It might pay to occasionally use such arrows in two or three dimensions, because always working in one (so effectively using only signed quantities) tends to obscure the vector nature of momentum.

Both mass and velocity

Wrong Track: This ball has the larger mass, so it'll have the larger momentum.

Right Lines: Momentum, a quantity that's a good measure of the motion of a body, depends on mass and on velocity. To work out which has the larger momentum you have to pay attention to both the mass and the velocity: →p = m × →v.

A focus on compensation

Thinking about the learning

Students often fix on only one factor, when they should consider both factors that contribute to a physical quantity. They select either mass or velocity, and they fail to combine the two quantities into the compound quantity that is momentum. So their comparisons are often in error.

Thinking about the teaching

The key relationship is: →p = m × →v.

That this kind of pattern (A = B × C) turns up so often, and so often causes troubles is one reason for focusing on the issue of compensation throughout the SPT materials. Because it's an often used pattern, and because students are known to have difficulty with it (through selecting just one of the factors) we'd suggest foregrounding it as a pattern, so that students give it due prominence.

Move between descriptions

Students ought to be developing a repertoire of descriptions for objects in motion, and relating changes in these motions to the forces acting on the objects.

Measures of motion:

Δ energy in the kinetic store = 12mv²

→p = m × →v

Changing the motion:

Δ →p = →F ×  Δ t

Δ (energy in kinetic store) = force × distance

→a = →Fmass

Teacher Tip: Practice translating from one representation to another, using the different measures of motion and changes in motion (momentum, mass, acceleration), and using different modes of representation for these different quantities (verbal, algebraic, graphical, and diagrammatic).

Two separate collisions: one over a longer duration than the other

Wrong Track: The force is spread out over a longer time, and over a greater distance.

Right Lines: There's a lot of momentum to remove from the moving body as the body is brought to a halt. The time over which the force acts is longer, so the force exerted is smaller. There's a lot of energy to shift from the kinetic store as the body is brought to a halt. The distance over which the force acts is greater, so the force exerted is less.

A focus on compensation

Thinking about the learning

It's a common pattern of reasoning amongst students, when faced with two physical quantities that affect a third, to foreground one of the pair and reason about changes to the third based on this chosen quantity alone. This is the difficulty common to focusing on changing either kind of measure of motion (energy in the kinetic store or the momentum of the moving object.)

There's another, quite separate, difficulty, that arises from the use of the phrase the force. Underlying this is often a tacit assumption that it's the same force in both the shorter duration collision and the longer duration collision. The interactions in the two collisions are rather different in magnitude so the forces that represent these two interactions will also be different.

It might seem a small point, but anything that leads to careful consideration of the two situations from first principles, and then compares them, is more likely to support meaningful learning than a rather careless phrase that could be used as a prop for an inappropriate short-cut. In particular, forces are not the kind of things that can be spread out. This kind of talking and analysis, where the force is worked on as one moves from one situation to the other, tends to lead off down the wrong tracks.

Thinking about the teaching

There are several representations in the Physics Narrative that support paying equal attention to both force and time or force and distance. We'd suggest using them, as well as the algebraic and verbal modes of reasoning.

Force and time are compensated quantities ( Δ momentum = force × time).

Force and distance are compensated quantities ( Δ energy = force × distance).

To counter the second difficulty, we'd suggest that you analyse the two collisions as two separate processes, and only then compare the magnitudes of the forces at the end.

Teacher Tip: Avoid referring to the force without qualification when comparing two collisions of different durations.

A hockey ball hits a well-padded goalkeeper: less pain results than for a similar strike on an unpadded goalkeeper

Wrong Track: The force is spread out over a larger area, so the head hurts less.

Right Lines: If the ball hits an unprotected body, only a few square centimetres of the surface interact with the ball on impact. Each square centimetre of the ball exerts a large force. When the ball hits the padding, many more square centimetres of the body interact with the ball than in a collision with no padding. More forces now act on the ball, from more square centimetres. Careful engineering of the pads results in each force being smaller.

Perform two separate analyses (with padding, and without). Then compare the findings.

Thinking about the learning

The difficulty here is that there are two separate collisions (one with padding and one without), and they need to be modelled separately, then compared. The temptation is to try to deal with a superposition of both, as a kind of short-cut. That's not a good plan until students are confident using the ideas, and maybe not even then.

Thinking about the teaching

Analyse one collision (without padding). Then analyse the second (with padding). Draw two separate diagrams, one for each analysis. Then compare the analyses. We think that a slower, more thorough comparison is more likely to give a secure understanding of what is being asserted and will set a better pattern for reasoning about new situations.

Two separate analyses

Here is a schema to follow:

Teacher Tip: A force replaces an interaction, so model the interactions first, and then introduce the forces later. Try to avoid talking about the force being spread out, as this could get messy.

There's a good chance you could improve your teaching if you were to:

Try these

• keep vectors and scalars separate
• emphasise common patterns
• allow the ideas to gain meaning by using them

Teacher Tip: Work through the Physics Narrative to find these lines of thinking worked out and then look in the Teaching Approaches for some examples of activities.

Avoid these

• defining quantities rather than putting them to use

Teacher Tip: These difficulties are distilled from: the research findings; the practice of well-connected teachers with expertise; issues intrinsic to representing the physics well.