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Consider the situation shown below.
A and B are initially at relative rest. B then accelerates away from A using a constant force.
After time t (as measured by B), B meets C and for a short time B stops the rocket motor so as to move with zero velocity relative to C.
During this short time, A measures the velocity of B (and C), and finds it to be 0·5c.
Now, let us transform C’s observations to A’s reference frame.
Velocity of B relative to A is v’.
Therefore, during the first period of acceleration, the velocity of B relative to A changed by 0·5c. During the second period of acceleration, it only changed by 0·3c but we have stated that the force causing the acceleration was constant.
We must conclude that from A’s point of view, the inertia of B steadily increased.
The mass of a body is a relative concept.
The mass of a body measured by an observer at rest relative to the body is called (not surprisingly) the rest mass of the body.
The mass (sometimes called the relativistic mass) of the body measured by other observers depends on the velocity of the observer relative to the body.
As the variation of mass is basically due to the time dilation effect, you should not be surprised to find that, if the rest mass of a body is mo, then its mass, m, as measured by an observer moving with speed v relative to the body is given by
As in Newtonian mechanics, the momentum of a body is often useful in the solution of problems. Momentum is the product of mass and velocity so, relativistic momentum is given by
In other words, relativistic momentum is calculated by simply taking the Newtonian definition of momentum but remembering that mass is now a variable quantity.
This can not be done with the kinetic energy of a body.
Mass and Energy
If a force causes body B to accelerate away from observer A (as in the example above) then work is done by that force. As usual we can define the kinetic energy possessed by body B (as measured by A) by saying that
It can be shown that if the relative speed of body B is such that its mass (as measured by A) is m, then
Here we are seeing the equivalence of mass and energy. Some of the work done by the force is converted into mass and if we define the total energy, E, possessed by a body to be the sum of its rest energy (moc2) and its K.E. we have the famous result
Units of Mass/Energy
The S.I. unit for mass is the kg and for energy, the Joule. However, on the scale of subatomic particles, we often use MeV for energy (1MeV = 1·6×10-13J). So, a possible unit for the quantity "mc2" is MeV.
For this reason, the masses of subatomic particles are often expressed in MeV/c2.
For example the rest energy of an electron is about 511MeV and its mass is therefore said to be 511MeV/c2.