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# Kinetic Energy of Relativistic Particles by Shana Priwer & Cynthia Phillips, Ph.D.

Remember the previous discussion of the kinetic energy of a particle as related to the particle's mass and its velocity in the equation E = ﬁmv2? This model works just fine for particles moving at slow, everyday speeds, but things become a bit strange when the speeds approach the speed of light. At speeds close to the speed of light, as energy is increased slightly, the mass increases along with the increase in velocity.

When speeds get very near the speed of light, they can't increase any more. Once particles hit this point, any increase in energy goes directly to an increase in mass of the particle! If we have a particle moving at a speed close to the speed of light, and we apply a force to it for a time interval of one second, the energy and therefore the mass of the particle will increase slightly, by an amount we can call m. Since the force is equal to the rate of change of mass, multiplied by the velocity, this gives us the equation F=mc (where F is the force, m is the slight increase in mass, and c is the speed of light, as usual).

## Increase in Kinetic Energy

So what is the increase in kinetic energy of the particle as a result of applying this force for one second? Remember that energy is the ability to do work, so the increase in energy is the work done during one second. The work done by the force is equal to the force multiplied by the distance. If the particle is traveling at the speed of light c, 186,300 miles per second, then in one second the particle travels 186,300 miles, or c miles. Therefore, the increase in kinetic energy of the particle is equal to the force multiplied by c miles.

So what is the end result? F=mc, and E=Fc. So we can combine them to show that E = mc * c = mc2! Does this look familiar? We have just shown where Einstein's famous equation E=mc2 comes from. It allows us to compare the increase in the mass of a particle moving at a speed close to the speed of light with its increase in kinetic energy.

## Slow Particles

Particles moving at speeds close to the speed of light increase in mass as compared to their resting masses. What about particles moving at more mundane, everyday speeds? It turns out that mass increases also take place for particles moving at much slower speeds. In fact, over the whole span of speeds from very slow to near the speed of light, particles experience a mass increase that's related to their increase in kinetic energy by the equation E=mc2.

So why don't we notice this effect in our everyday life? Does your mass increase when you are running as opposed to when you are standing still? Even though it may feel like it does, the increase, while real, is so tiny it is very difficult to measure. This is true even at speeds that are significant but still much less than the speed of light. For example, the mass increase of a typical airplane flying at 2,000 miles per hour would only be about half a milligram as compared to its mass while at rest on the ground! This amount is almost undetectable.