Suppose the speed of light were 100 mph. Trains on parallel tracks are approaching each
other at speeds of 92.64 mph and 87.89 mph. A person on one of the trains sees the two
trains approaching each other at what speed? Answer in units of mph.
After the trains pass a crazy person on the 92.64 mph train fires a rifle with muzzle velocity 92.64 mph back at the other train. With what speed does the person on that
train see the bullet approach? (A negative answer means the bullet is receding).
Answer in mph.
In c = 1 units the usual formulae are still valid. To convert the speeds to these units all you need to do is divide the speeds by the value of the speed of light in the original units. So, the (dimensionless) speeds in the c=1 units are just v1 = 0.9264 and v2= 0.8789. At the end of the calculation you just convert back to the original units by multiplying the speeds in c = 1 units by the value of c in the original units to obtain the value of the speeds in the original units.
You can "add up" the speeds using the formula:
v_rel = (v1 + v2)/(1+v1*v2/c^2)
in c = 1 units this is just:
v_rel = (v1 + v2)/(1+v1*v2)
Here positive v_rel means the trains are approaching each other. See below for the derivation.
You can just substitute c = 1 in the equation, or you can divide v_rel by c to convert that to c = 1 units:
v_rel/c = (v1/c + v2/c)/(1+v1*v2/c^2) =
(v1/c + v2/c)/[1+(v1/c)(v2/c)] ...
A detailed solution is given.
Relativistic Physics Understanding
A spaceship travels with a speed of 0.6 c as
it passes by the Earth on its way to a distant
star, as shown in the diagram below. The
pilot of the spaceship measures the length of
the moving ship as 10 m.
Determine its length as measured by a per-
son on Earth. Answer in units of m.
The pilot of the spaceship observes that the
spaceship travels for 2 years.
Determine how much time has passed ac-
cording to a person on Earth. Answer in units
Some time after passing the Earth, the pilot
shoots a laser pulse backward at a speed of
3 *10 ^8 m/s with respect to the spaceship.
Determine the speed of the laser pulse as
measured by a person on Earth.
Answer in units of m/s.