The Quantity That Survives a Collision
Last timeTwo Kinds of Energy, One Total
Energy can be lost to heat, so it is the wrong thing to hold a collision to. Momentum cannot be lost at all, and a calculation can keep it exactly.
A collision is the hardest thing in mechanics to describe and the easiest thing
to solve. Hard, because what happens during the contact involves the insides of
both bodies deforming, heating and ringing, and nobody wants to model that.
Easy, because there is a quantity that does not care.
- how much there is of body i, which never changes during the collision
- how fast body i is moving, with its sign carrying the direction, so two bodies moving toward each other partly cancel
- the total, which is a single number for the whole system in one dimension and one arrow in general
The reason this particular combination is conserved is worth stating plainly,
because it is not a coincidence and it is not about collisions. Whatever force
body one exerts on body two, body two exerts the opposite on body one. So in the
sum above, every force inside the system shows up twice with opposite signs. The
total can only be changed by something reaching in from outside.
Exact, not nearly
That cancellation is not an approximation, and it survives being written as a
stepped calculation. Below are two carts connected by a spring, pushed together
and let go. The carts weigh one and three. Their velocities do whatever they
like; the total is held to the last digit.
The lesson stops here
6 more paragraphs to go
You have read the opening. The rest of the argument, the problems that check whether it landed, and the lines worth keeping at the end all come with a plan.
The first lesson of every course in the library reads the whole way through, free, so you can see exactly what the rest of them are.
See the planThe contentsThis is the reading half
Starting the course gives you your own copy of it. Every idea on every page has problems standing under it, marked with a reason rather than a tick, and any sentence you do not believe can be opened and argued with. None of that can happen on a page nobody owns.
The contents