Rosencrantz (an anguished cry): CONSISTENCY IS ALL I ASK!
Tom Stoppard, Rosencrantz and Guildenstern Are Dead (1966)
I think that dual coding techniques can be extremely helpful in helping students understand the concept of change of momentum.
To engage our students’ physical intuitions, let’s consider a question like: Which would hurt more — being hit by a sandbag or being hit by a rubber ball?
Let’s assume that the sandbag and rubber ball have the same mass m and are travelling at the same initial velocity u. We choose ‘u‘ because it’s the initial velocity and we take ‘v‘ as the final velocity: a very subtle piece of dual coding that can reap rewards if applied consistently — pace Rosencrantz(!) — over a range of disparate examples.
To analyse this problem, let’s use the momentum version of Newton’s Second Law of Motion.
We will use the change = final – initial convention (‘Consistency is all I ask!’)). The initial momentum is pi and the final momentum is pf.
Now let’s work out the change in momentum in each case. We will assume that each item is dropped so that it impacts vertically on a horizontal surface. The velocity just before it hits is u so its initial momentum pi is given by pi = mu; its final velocity is v so its final momentum pf is given by pf = mv. The sandbag does not rebound, so its final velocity v is zero.
The rubber ball rebounds from the surface with a velocity v (we have shown that v < u so we are not assuming a perfectly elastic collision).
We will use the down-is-positive convention so that u is positive and the downward momentum pi are positive in both cases. However, the velocity v of the ball is negative so the momentum pf = mv is negative (upwards).
To add vectors, we simply put them ‘nose to tail’. However, in this case, we need to subtract the vectors, not add them. To do this, we use the operation pf + (-pi,). In other words, we put the vector pf nose to tail with minus pi, or with a vector pointing in the opposite direction to the original vector pi. These are shown in the table.
We can see that the change in momentum Δp is larger in the case of the rubber ball.
Applying Newton Second Law that force = change in momentum / change in time then (assuming the time of each interaction is the same) then we can conclude that the (upward) force exerted by the surface on the ball is larger than the force exerted by the surface on the sandbag.
From Newton’s Third Law (that if an object A exerts a force on object B, then object B exerts an equal and opposite force on object A), we can also conclude that the rubber exerts a larger downward force on the surface. This implies that, if the ball hit (say) your hand, then it would hurt more than the sandbag.
Considering change of momentum problems like this helps students answer questions such as the one shown below:
We can discard options C and D since the change of momentum shown is in the wrong direction: the vertical component of momentum will remain unchanged.
A and B show changes of momentum of the same magnitude in the horizontal direction. However, if we take the horizontal component of the initial momentum as positive then the change of momentum on the gas particle must be negative; this implies that the correct answer is B.
Note also that diagram B shows the pf + (-pi) operation outlined above, with the arrow showing minus pi shown in red (added to the original exam question).
e=mc2andallthat 
This story about bouncy thing hurts more than non bouncy thing always strikes me as odd. I do understand the approach of making an exercise in equation wrangling relate to „everyday experience“. But, to come to my point quickly, the assumption of equal contact time, which is made so casually and without any further comment is of course entirely arbitrary and hugely unlikely in actual „everyday experience“.
I work at a school as a technician and find teachers respond with sheepish evasion to such questioning. Since you write more extensively about the process of science education, I hope I can elicit a deeper response.
The other casual, and in most cases so far off the mark it can be called misleading, simplification lies (hidden?) in talking about „the force“. Force, i.e. the intensity of the interaction between ball and hand is and can only be not constant over the time of contact. Just try to imagine something that deforms elastically with force remaining constant as deformation becomes deeper!
When you teach crumple zones, do you not talk about maximum force, as contrasting average force, as something which needs to be kept as small as possible because it is what determines damage to occupants?
There is a wider aspect to this nitpicking: Simplifying assumptions are of course necessary to turn reality into a model which is simple enough to be operable by teenagers with limited attention spans. But if these simplifications are not continuously made explicit they are forgotten by the teachers. I have worked long enough in an independent school where all teachers in our department have physics degrees to speak from experience.
A more publically visible example is the assumption in a required practical on density. AQA instructs students to assume that adding salt to water does not change the volume of liquid. Verification that this is a nonsensical assumption ( albeit necessary to produce a simple calculation) is easy and quick to do, but oddly refused to address by nearly all teachers I know.
In conclusion, let me frame the question this way: is there a better way to handle the conflicting demands of „reality“ and operable model, than this detached folklore of teachers stories?
Thanks for the comments — you make some really interesting points. To my shame, I can definitely plead guilty to the charge of sloppily using “force” instead of “average force” when discussing the rate of change of momentum. However, in partial mitigation this is often when covering the topic at GCSE where, due to scheduling, it has to be taught during a single lesson. I would like to think that I am considerably more careful when dealing with the topic at A-level; but even then, talking about ‘the rate of change of a rate of change of a rate of change’ tends to make even well-motivated students’ eyes glaze over. Your point about crumple zones is well taken, and it’s the maximum force that is, of course, the killer (possibly literally). The tension between making a treatment comprehensible to the students in front of us while maintaining strict scientific accuracy is a circle that is, at least in my experience, is impossible to square. Your point that we should, at the very least, acknowledge the simplifications that are part of the shared ‘detached folklore of teacher stories’ (great phrase, by the way) that most teachers (myself included most of the time) are unaware of in the same way that we assume that fish are unaware of the water they swim through. I was aware that my taking of the contact time between an elastic and inelastic collision as constant was a simplification but hadn’t given much thought to whether it was a reasonable assumption . . . You have given me a lot of food for thought — thank you!