There was a question posted some time ago regarding the
of temperature. The answer given was that there is no limit to
temperature beacuse it is a measure of the average kinetic energy
of particles,but what if the average kinetic energy of those particles
approached the speed of light (natures speed limit according to the
theory of relativity) wouldn't the temperature eventually reach
an enormous yet FINITE limit, as the kinetic energy of the
particles approached, but could not exceed, the speed of light?
The speed of light does not limit the kinetic energy of particles. In
the usual way of looking at it, as you approach the speed of light the
amount of energy required to go faster increases without limit.
However, the argument relating temperature to kinetic energy is gounded
in nonrelativistic physics, and there are other ways of looking at
temperature. In statistical mechanics, you look at the number of
different states available to a bunch of particles, and temperature
describes how this number changes as the total energy increases. From
this view, temperature can even be negative, so there certainly is the
possibility of a maximum temperature.
The kinetic energy has two components: mass and velocity. As velocities
approach the speed of light the mass starts to increase, thus requiring
more energy to continue to accelerate the object. While the speed of light
limits the velocity, the mass can increase indefinitely, thus the kinetic
energy continues to increase.
No, there is no upper limit. While there is a finite limit to the velocity
the particles may travel at (the speed of light), the kinetic energy may in
fact become infinitely large.
At low speeds, the kinetic energy may be approximated by the equation:
K = 0.5 * m * v * v
However, at near light speeds, the correct equation should be used:
K = m * c * c * (1 / (sqrt(1-(v/c)*(v/c)) - 1))
In this equation, as v becomes closer to c, the speed of light, the equation
approaches infinity. When v = c, the exact value is undefined since 1 is
divided by 0, but it is some infinitely large number.
Basically, think of it this way, kinetic energy depends on the velocity and
mass of the particle. As the speed of the particle approaches the speed of
light, the mass of the particle also increases until it is infinitely large.
At low speeds, the change in mass is so small it is largely ignored and has
no noticeable effect on the results. However, at near light speeds, just
the opposite occurs. The change in velocity is less important to the final
result than the change in mass.
Speed and kinetic energy are related, but they are not the same thing. It
is incorrect and in fact misleading to think of the kinetic energy of an
object approaching the speed of light.
Although it is true that no object with mass can travel at the speed of
light, that doesn't mean that there is an upper limit to an object's kinetic
energy. This is because an object's mass increases as it travels faster.
In fact, by the theory of relativity, an object traveling at the speed of
light would have infinite mass. In terms of kinetic energy, what this means
is that the kinetic energy of an object, given by the formula mv^2,
increases faster than just the square of the velocity. As v approaches c, m
approaches infinity, and so does the kinetic energy.
Richard E. Barrans Jr., Ph.D.
PG Research Foundation, Darien, Illinois
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Update: June 2012