Energy Momentum Relation
There is a way to relate momentum and energy that proves useful in certain contexts. While total energy depends on kinetic energy and therefore velocity, the momentum accounts for the velocity as well. In this sense it aught to be possible to express total energy as some combination of rest energy and momentum. I will show the relation and prove it is equivalent to the total energy we saw in the last section.
Energy Momentum Relation
The relation is shown below in the first line. Following are the steps to show that it is in fact equivalent to the total energy from the last section:
As you can see, this is the same expression for total energy that we had in the last section.
EXAMPLE: An electron has a total energy of 2.0MeV. What is its kinetic energy and momentum? The mass of an electron is 0.511MeV/c2.
SOLUTION: The kinetic energy is the difference between its total energy and its rest energy. The rest energy is and the mass is given. Therefore the rest energy is just 0.511MeV. That leaves K=2.0MeV-0.511MeV = 1.489MeV.
The electron's mass is 0.511MeV/c2. Plugging into the energy/momentum relation gives us
Notice that this electron is traveling rather fast such that its kinetic energy is K=(2.0-0.511)MeV=1.489MeV, or roughly three times its rest energy. This will never be the case for macroscopic particles.
Ultra-Relativistic Limit
Notice that the energy/momentum relation can be seen like the Pythagorean theorem where the sides of the triangle represent the terms. The hypotenuse represents total energy while the two orthogonal sides represent the momentum and the rest energy of the particle.
In the event that the particle is moving a very large fraction of the speed of light, the momentum will dominate over the rest energy. In this case we may approximate the total energy (hypotenuse) as being equal to the long side of the triangle (momentum times c) of the particle.
Thus E=pc is true for both light which doesn't have rest mass, and for massive particles when traveling at very large speeds with large Lorentz factors
EXAMPLE: An electron is accelerated across a 1MV potential. What is its total energy?
SOLUTION: The electron acquires 1MeV of kinetic energy after accelerating across the potential. This means total energy is
EXAMPLE: What is the speed of that electron?
SOLUTION: Since we know the rest and total energies, and since we find This leads to
EXAMPLE: What is the electron's momentum if it's moving in the x-direction?
SOLUTION: There are two options to calculate momentum using what we know. Either plug directly into or use the energy/momentum relation and plug in the total energy and rest energies to get which leads to the same answer for momentum.
EXAMPLE: Two particles of mass m collide head-on while traveling at 0.6c. They combine to form a new particle. What is the mass of the new particle?
SOLUTION: Initially the two particles have both kinetic and rest energies, or each with total energy given by Based on a velocity of v=0.6c we find first that This means total initial two-particle system is The collision must conserve momentum as well as energy. If the particles are traveling in opposite directions at 0.6c, then Therefore we can conclude that final momentum and the final velocity are both zero. So the resulting particle will only have rest energy (no kinetic energy). Equating initial and final energies gives