RELATIVISTIC MOMENTUM AND KINETIC ENERGY

The convention now in physics is to say that the mass m of a body is the measured value when the body is at rest with respect to the measuring device. The subscript has been dropped from the m_o for convenience. However, proper length L_o and proper time t_o are still in use, so I will stick with m_o. If we use f = ma or p = mv, we will find that the m to use is gm_o. This is not the conventional jargon, but it does the job.

We find through relativity theory that speed c cannot be achieved by a mass. Therefore something must be happening as v increases to reduce a force's ability to cause an acceleration. An obvious possibility: mass increase. We speculate that the relativistic mass is m = g m_o. This would do the job, since gamma goes to infinity as velocity approaches c. Now let us check this. Consider a body which has momentum p. This should be relativistic mass m times velocity. Remember that force is dp/dt. Exert a force to speed it up over a distance dx. Then the work done is the change of energy, or (dp/dt)dx = c² dm. The left-hand side is (dp/dv)vdv after playing with the chain rule. (Do it.) The right-hand side is c² mdg . Now show that both sides reduce to

m_ovdv(1 - v²/c²)^-1.5

where m_o is the rest mass. Thus our equations are consistent. Here is another way to approach this. We have from above,

c²dm = (dp/dv)vdv

For p we use mv, so dp/dv is m+vdm/dv. Getting all m terms on the left, this becomes

dm/m = vdv/(c² - v²). Now let u = c² – v² so du = -2vdv

Show the above, then using m=m_o at v=0, show that

ln(m/m_o) = ln{[(c²-v²)/c²]^-0.5 }

lna = lnb à a=b, which means that m = g m_o.

When you speed a body up, the energy you give it results in added mass. Thus kinetic energy is found by K = (m-m_o)c² , or m_o(g-1)c²

Again, remember that the standard terminology today is to use m for rest mass, not m_o, and use gm if it is moving fast.

Whenever you add energy of any kind to a body, you are adding mass to it. Of course the thing which gave up the energy has lost mass, and the total mass-energy of the universe is constant. (There are theories about mass traveling through "wormholes" to other universes, so maybe the mass-energy of our universe is not constant. Some theories which sounded crazy at first have been found to be true. Others remain destined for the loony bin.) Back to main physics page or derive E = mc^2.

A handy relationship between momentum p and kinetic energy K is (pc)² = K² + 2Km_oc² which leads to E² = (pc)² + (m_oc²)². If you draw a right triangle with E for the hypotenuse, m_oc² for the side adjacent to q and pc the side opposite, then sinq = v/c and cosq = 1/g. The triangle has no significance other than as a mnemonic.

My other main pages:

fluids, heat, electricity and magnetism
vibrations and waves
quantum
index

Comments, questions: fredrick.gram @ tri-c.edu (remove spaces)