Since I didn’t want to hijack another thread…
Is energy and momentum the same thing?
How about momentum and inertia?
Energy and momentum are related in the same way as length and time, basically; they aren’t identical, but they are similar.
To give the mathematics of it:
momentum = mass * velocity
energy = 1/2 (mass * velocity[sup]2[/sup])
Well, really, to give the mathematics the way I meant it, we mix length and time by transforming (ct,x) using the good old Lorentz transformation, which we all know and love from beginning relativity. The important thing is that we can mix space and time with an awful lot of flexibility as long as c[sup]2[/sup]t[sup]2[/sup] - x[sup]2[/sup] doesn’t change.
In the same way, we can mix energy and momentum by mixing (E/c,p) with Lorentz transformations; we maintain E[sup]2[/sup] - p[sup]2[/sup]c[sup]2[/sup] = m[sup]2[/sup]c[sup]4[/sup] in doing this.
But that was more equations than I think we really needed; the point is that energy and momentum are two sides of the same coin in the same sense as length and time are two sides of the same coin. The equations wolfmeister quoted are essentially what we would get if we pretended relativity was too inconvenient so could be ignored.
Relativity is too inconvenient and can be ignored, in the vast majority of situations. The Newtonian equations Mr. Meister gave were perfectly adequate for demonstrating the similarity between momentum and kinetic energy.
Conservation of momentum is basically Newton’s law of action and reaction. If I’m standing on a frictionless surface and push someone away at 1 meter/sec, I’ll end up moving at 1 mph in the opposite direction because the total momentum of the system (two people) is conserved. Before the push, each has zero speed and, say, 70 kg mass so each has zero momentum. After the push, he has [70 kg * 1 m/s = 70 kg-m/s] of momentum in one direction and I have -70 kg-m/s (minus means opposite direction) so the total momentum is zero, and is conserved. But note that energy was not conserved in this action - I did work and added kinetic energy to the system. Before the push, kinetic energy is zero. After the push, each of us has positive kinetic energy (1/2 * 70 kg * (1 m/s)[sup]2[/sup] = 35 kg m[sup]2[/sup] s[sup]-2[/sup] = 35 Joule).
if two perfectly elastic objects collide how do you predict where they will bounce ? considering either momentum or energy is not enough, it is only when you realize that both momentum and energy have to be conserved then you can accurately predict it.
for example. if one elastic ball is stationary and another hits it, how do you know if they now both travel in same direction or the moving ball bounced back ? intuitively you know it will bounce back if it was lighter, but the mathematics of it is in the relationship of momentum and energy.
energy is not a vector, it has no direction. upon bouncing off a perfectly elastic wall the momentum reverses but energy remains the same.
a light, fast-moving particle is almost all energy, very little momentum ( because speed comes in square with energy ). after elastic impact the system must conserve that energy, thus something must still be moving fast after the impact, and there is not enough momentum to move the heavy particle fast, so it will be the light one moving fast still. but it can’t move faster than the heavy particle in the same direction unless it penetrates right through (which would not be an elastic collision) - so you know it bounced back.
oh and, the total system momentum is still very small after collision because now the particles are moving in opposite directions partially cancelling each other’s momentum, but adding in energy.
I disagree. It illustrates a mathematical relation under a particular set of assumptions; it doesn’t show the essential relation between energy and momentum. I recognize that relativity can be ignored for most purposes (I am, after all, a practicing physicist myself), but that’s no reason to pretend that it can always be ignored. Especially if, as I expect, the reason this came up to begin with had to do with the “energy and momentum of the universe” thread, in which case we certainly can not pretend that relativity is negligible.
The nonrelativistic equations for momentum and energy do not show the true similarities of the two quantities. That they involve mass and velocity, yes; but that’s hardly the extent of it.
Conservation of momentum is a result of the translational symmetry of space (i.e., the fact space looks the same everywhere). Conservation of energy is a result of the translational symmetry of time (i.e., the fact that time looks the same everywhen). Momentum-energy (or “momenergy” as my GenRel prof used to say) is thus an intrinsic property of all particles in the space-time, resulting from the nature of space-time itself. This is why it’s at all useful to talk about momentum and energy.
Although momentum-energy is a single quantity in GenRel, going back to Newtonian physics we have to deal with the two separately. This greatly simplifies things for the most part, but we lose the symmetries between the two. (For example, v vs v[sup]2[/sup].)