Question

You're on a highly-classified project to design rockets of incredible speeds. There are two situations, one is a two-stage rocket and one is a three-stage rocket.

The two-stage rocket reaches a speed of .5c in relation to the launch-pad in deep space. In relation to the first stage, the second stage reaches a speed of .5c.

Each stage in the three-stage rocket reaches a speed of (1/3)c in relation to the launch-pad or in relation to the stage it separated from.

Calculate the final speeds of the last stages, and determine which rocket has the highest final speed. (c = speed of light)

Answer 1

Lets work for an n-stage rocket, so that the relative speed w.r.t. previous stage is c/n.

Work in terms of rapidity, because then rapidities add like Galilean transformations. Rapidity is defined as

where

So, the "relative" rapidity of each stage is

The final rapidity, i.e. of n-th stage is

Convert this back into speed

After this its just mathematics,

For n = 2,

For n = 3,

Thus,

In fact,

For mathematicians, the case , may be of interest.

There is nothing advanced about my method. It is just convenient mathematically. According to Lorentz transformation the relativistic velocity addition is (this you probably know):-

Divide both side by c and substitute

for rapidity x,y,z corresponding to u,v,V. You will get the result I was talking about. I used the following identity for the above result.