Momentum and Energy?

Question

so my teacher is testing us on these two topics...
need an any length of explaination of energy and how it relates and differs from momentum...
i really need to be tutored on energy...but my teacher doesnt teach us too well

Answer 1

OK, I'm presuming you're not a postgrad physics student. So you really don't want to hear about Hamiltonians and other arcane matrices and operators. What you want is plain vanilla, common knowledge about momentum and energy.

First, energy...energy is the capability to do work or cause a change. That's all it is. In fact one can say Work = Energy = Force X distance moved by the Force = Fd. So energy (work) units are Newton-meters, which also has an arcane name called a Joule. But I prefer the standard metrics kg-m^2/sec^2.

Second, momentum...momentum is just a name we give to mass times velocity of that mass = Mv. The momentum units are kg-m/sec. If you compare these units with the energy units above, you see how closely related they are...momentum only lacks a meter (m) unit to have the same units as energy.

Now here's where they are related. The change in momentum over time = del(Mv)/delt = F = Force; where del means "change in," Mv is momentum, and delt is the change in time. Let me repeat: the change in momentum over time is Force; so that del(Mv)/delt = F.

So, for example, if you want to know how much work (energy) is needed to move a mass (M) d meters, we can use Energy = Fd = (del(Mv)/delt) d = M(delv/delt) d = Mad; where M is the mass moved, a is the average acceleration of that mass as it is pushed with Force F, and d is the distance the mass traveled with average acceleration a.

del(Mv) = M delv because M (mass) does not change when the force (F) is applied to it...but its velocity (v) does; so we have M delv/delt and because delv/delt = average acceleration = a; we find that the change of momentum over time is Ma and we have the famous F = Ma equation you've probably seen. That's right, the Ma part of F = Ma is the change of momentum over time.

So there you have it. Work = Energy = Fd. But F = del(Mv)/delt which is the change of momentum over time. So Energy = the change in momentum X the distance the mass traveled = Mad.

Answer 2

Let me start off with the formulas for computing the momentum and energy of a moving object. Let m = mass of the object, v = velocity, p = momentum, and E = energy. The formulas are:

p = m v
E = 1/'2 m v^2

So, they are not the same thing, but you need both in order to predict what will happen should the object hit another object, like what happens with balls on a pool table, especially if the balls don't all have the same mass. If we assume that both momentum and energy are conserved, then we can determine the outcome, which is unique. Force exerted on a moving object multiplied by time it's exerted will give you the change in its momentum. Force exerted on a moving object multiplied by distance it's exerted will give you the change in its energy. This can be expressed by the following:

F * dt = dp
F * dx = dE

where d = delta, or change in

In physics, for every physical symmetry, there is a conserved quantity (see Noether's Theorem). If the outcome of a collision is independent of position in space (translational symmetry), then it implies conservation of momentum. If the outcome of a collison is independent of time (temporal symmetry), then it implies conservation of energy. These are deep concepts, and lead to both Lagrangian and Hamiltonian interpretations of physics. In Quantum physics, x and p, that is to say, position and momentum, are conjugate variables, and t and E, that is to say, time and energy, are also conjugate variables, so that a quantum state function decribing one, like p, will describe the other, x, through the Fourier transform. Finally, in relativity theory, the metric is based on x and t, that is to say, position and time, and its counterpart is based on p and E, that is to say, momentum and energy, and Einstein's OTHER famous equation, G = 8 pi T, relates the two, G being the Riemannian curvature tensor of spacetime, which involves just x and t, and T being the Stress-Energy tensor, which involves p and E. So, we do need to have both p and E.

Answer 3

Energy = mass X (speed of light)^2

momentum = velocity X mass

-----------------
mass = Energy / c^2

mass = momentum / velocity

so

Energy / c^2 = momentum / velocity

and -

Energy = momentum X c^2 / velocity

momentum = Energy X velocity / c^2