Question

A spaceship at rest in a certain reference frame S is given a speed increment of 0.50c. Relative to its new rest frame, it is then given a further 0.50c increment. This process is continued until its speed with respect to its original frame S exceeds 0.999c. How many increments does this process require?

Short answer

A minimum of 7 increments would require to exceed 0.999c.

Step-by-step solution

Relativistic velocity addition

Suppose an object is moving with the velocity $u\text{'}$with respect to S’ frame, which is moving with the velocity $v$with respect to frame Sthen the velocity of the object $u$with respect to S frame is

$u=\frac{u\text{'}+v}{1+\frac{u\text{'}v}{c^2}}$

Determine the number of increments required.

In the first increment, the spaceship moves with $u_1=0.5c$with respect to the S frame. In this new rest frame, a further 2^nd increment of v = 0.5c is given. The velocity relative to the S frame is

$$\begin{gathered} u_2=\frac{u_1+v}{1+\frac{u_{1}v}{c^2}} \\ =\frac{0.5c+0.5c}{1+\frac{0.25c^2}{c^2}} \\ =\frac{1c}{1.25} \\ =0.8c \end{gathered}$$

This process is repeated; in the new obtained rest frame, the 3^rd increment of 0.5c is given. Then the velocity relative to the S frame will be