An object moving to the right at 0.8c is struck head-on by a photon of wavelength lambda, moving to the left. The object absorbs the photon (i.e. the photon disappears) and is afterward moving to the right at 0.6c.
(a) Determine the ratio of the object's mass after the collision to its mass before the collision.
(b) Does kinetic energy increase or decrease?
It is convenient to use c = 1 units. We are going to evaluate the ratio of the masses and the ratio of the kinetic energies which are dimensionless, so they take the same values in any system of units. Velocities are dimensionless in c = 1 units. The numerical value of a velocity in c = 1 units is equal to the ratio of the velocity and the speed of light in any other system of units.
Before we start, let's review some relativity theory, in particular four-momentum vectors. The four-momentum of a particle is given (in c = 1 units) by (E, Px, Py,Pz), where E is the energy and Px, Py, Pz are the three components of the momentum. The inner product between two four-vectors is defined as:
(V0, V1, V2, V3) dot (W0, W1, W2, W3) = V0 W0 - (V1 W1 + V2 W2 + V3 W3)
This definition makes the inner product between four-vectors invariant w.r.t. Lorentz transformations. We can derive the mass energy relation using this inner product as follows. Suppose a particle has an energy E and momentum P in one frame. In that frame the four-momentum vector is (E,P). The inner product of this vector with itself (in the following this will be called "square of the four-vector") is E^2 - P^2.
In some other ...
A detailed self-contained solution is given. Four-momentum algebra is first explained and then applied to the problem.