if the two events occur at the same location in the reference
frame of the observer. An object’s length is said to be proper if
the object is at rest in the reference frame of the observer.)Relativistic Energy
You’ve undoubtedly seen the formula E = mc^2 , probably the most famous formula in
physics. What it says is that mass (m) and energy (E) are equivalent, and the
formula tells us how much energy is equivalent to a given amount of mass. This
energy is called rest energy because an object resting, say, on your desk, has
energy—in fact, is energy—simply by virtue of the fact that it exists and has mass.
Because c^2 , the square of the speed of light, is such a big number, a small amount of
mass is equivalent to a huge amount of energy. When an exothermic nuclear reaction
takes place, the total mass of the product nuclei is always less than the total mass of
the original nuclei. The “missing mass,” ∆m, has been converted to energy, ∆E, in
accordance with the equation ∆E = (∆m)c^2. This is the basis for how the world’s
nuclear power plants generate energy. The fission of a neutron with a uranium-235
nucleus creates five product particles (a barium-140 nucleus, a krypton-93 nucleus,
and 3 neutrons) whose total mass is less than the total mass of the original neutron
and uranium-235 nucleus by about 3 × 10−28 kg. Multiplying this ∆m by the factor
c^2 = 9 × 10^16 m^2 /s^2 , we get ∆E = 2.7 × 10−11 J.
But what about kinetic energy? Up to now, we’ve been using the formula KE =
mv^2 for the kinetic energy of an object of mass m moving with speed v. But if v is
close to c (in which case we say that the object is moving at a relativistic speed),
then this formula won’t work. The correct formula is
KE = (γ−1)mc^2where γ is the usual relativistic factor. It can be shown that when v is very small
compared with c, this formula becomes the familiar KE = mv^2.
An object’s total energy, Etotal, is now defined as the sum of its rest