Momentum and energy in GCSE-level physics

Some things are obvious, yet never explicitly discussed. When it comes to GCSE physics, it's that there's some interesting relationship between momentum and energy. Something is clearly going on here. Momentum is mv, energy is 1/2 mv^2, energy looks like some kind of integral of momentum or something, yet we never discussed this at GCSE or A-level. Maybe it's broken dug into if you go beyond A-levels, but that where my Newtonian physics knowledge tops out, leaving it as something of a mystery.

Mind you, GCSE and A-level physics don't really try to explain anything (or at least they didn't for us) - the formulae for momentum and kinetic energy were given, and that was it. No justification for why they'd be logical quantities to care about.

As we'd concentrate on solving physics problems through the conservation of both momentum and energy, it took me quite a while to realise that Newton's laws of motion are just in terms of momentum, not energy. Mechanics problems can be solved without direct reference to energy!

I find this very interesting as reaching the same solutions with or without explicit use of conservation of energy implies that conservation of energy in mechanics is not so much an axiomatic law in itself as something that derives from how Newton's laws work. I suspect this is the kind of thing Emmy Noether knew all about, although I've never built up the physics maths to actually understand Noether's theorem.

It does rather feel like momentum does deserve primacy in mechanical physics. A simple quantity, linear in how heavy the object is, and how quickly it's moving. It transforms sensibly under reference frame, unlike kinetic energy, with that dodgy velocity-squared term.

This is odd, because in other areas of physics, energy is clearly king. Chemical energy, thermal energy, electrical energy. No-one talks about momentum in those situations. Why this is seems a little deeper than I can fathom.

Still, solving mechanical problems with forces and momentum alone, and no reference to energy, is a fun change of mindset compared to my A-level physics. I'm pretty sure it's the kind of thing anyone who's ever written a physics simulation will know inside out, but I'm happy to reinvent from scratch.

For example, I'd usually treat a ball bouncing elastically on a surface as a conservation of energy, flipping the velocity component perpendicular to the surface. Instead, we can create a strong repulsive conservative field on the surface to repel the ball. As the ball passes through the field, it loses its velocity component perpendicular to the field (as that's how we've set the force up), until it stops. And then, thanks to time symmetry (given an appropriate field), it'll accelerate up to the same speed as it entered, when it leaves the field. More complex problems can be analysed similarly.

Given that we can solve these problems without reference to "energy", but also that the concept of "energy" tends to make solving the problems simpler, "energy" clearly encapsulates some idea, but we still haven't really articulated what it is. What is it?

We can start with momentum. An object can gain momentum M if subjected to a uniform force F for t seconds, where M = Ft. So momentum is force integrated over time.

On the other hand, a uniform force F working on a mass m leads to a uniform acceleration of a, and with v = at, the distance travelled (from a standing start) at time t will be 1/2 a t^2 = 1/2 v^2 / a = 1/2 m v^2 / F. Put another way, if s is the distance travelled, energy E = Fs. Energy is force over distance and... oh, I appear to have re-derived the unexplained-at-GCSE/A-level "work is force times distance".

I'm not quite sure what this means, but momentum and energy appear to be the time and space versions of the same thing. We can see this with dimensional analysis - momentum in Ns and energy in Nm. The fact that energy is a space integral of force means that we can derive a (scalar) energy field over space, from an appropriately conservative force. I assume that, by symmetry, an appropriate force field can be integrated over time to give a field of momentum change.

When I look beyond A-level, to one of the few pieces or more advances physics I've studied since, special relativity, I see that momentum and energy are unified there, too. "Momenergy" is a 4-vector where the spatial dimensions are momentum and the time dimension is energy. When I think about this, it reveals something I'd missed: Momentum is a space-y, vector-y value, and energy is scalar. I'd thought of momentum as a time integral and energy as a space integral. What gives?

As we integrate force in both time (for momentum) and space (for energy), we can maybe view force as a derivative in time and space. Integrating over time leaves just the space component for momentum, while integrating over space leaves the time component for energy. The pair are related, but not quite as I thought.

At the end of all this stumbling around in the dark, I do rather wish that I'd done physics to a higher level. I don't think that it necessarily leads to a magical epihany on the connection between momentum and energy (with perhaps the exception of if you really understand relatvity!), but at least it would provide a larger toolbox with which to explore.

(Post started on 2024-01-07 - I'm a slow writer/editor these days!)

Posted 2024-01-15.