Question

Zosia flies her length L spaceship towards a length 2L space station, which has open doors on each end. How fast must Zosia fly her spaceship so that her ship appears exactly as long as the space station in her reference frame?

A. 0.99 c

B. 0.87 c

C. 0.50 c

D. 0.43 c

E. 0.25 c

Answer 1

If l0 is the proper length in rest frame and l denotes the contracted length in the frame of reference of the observer moving with velocity v then,

$l=l_0\sqrt{1-\frac{v^2}{c^2}}$

c is the velocity of light

For the observer in a spaceship, the contracted length must appear L as compared to the original length 2L

$\therefore L=2L\sqrt{1-\frac{v^2}{c^2}}$

$or,\ \sqrt{1-\frac{v^2}{c^2}}=\frac{1}{2}$

$or,\ \frac{v^2}{c^2}=1-\frac{1}{4}=\frac{3}{4}$

$or,\ v=\frac{\sqrt 3}{2}c=0.866c\approx 0.87c$

Option B is correct.