Relativity Lite

16 | Relativity Lite Now suppose the rocket is moving at velocity v = 3 5 c relative to the Earth. If the space- ship were at rest, light emitted from the left-hand mirror would have to travel a distance ℓ to the right-hand mirror in figure 9a. But in time t a , that right-hand mirror moves with the spaceship an additional distance, a = v t a , before the light catches up (figure 9c). So the distance that light must move in that same time is c t a = ℓ + v t a . Subtracting v t a from both sides, collecting terms, and using v = 3 5 c gives ct vt ct v c ct a a a a − = − = −⎛ ⎝ ⎞ ⎠ = ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ 1 1 3 5  , or c t a = 5 2 ℓ . In this time, the rocket travels 3/5 as far as the light, a = v t a = 3 5 c t a = 3 2 ℓ . A much shorter time later, t b , the reflected wave collides with the bottom left-hand mir- ror, traveling toward it rather than away this time (figure 9d), so that c t b = ℓ − v t b , or ct b vt b ct b v c ct b § © ¨ · ¹ ¸ § © ¨ · ¹ ¸ 1 1 3 5  , and (multiplying both sides by 5 8 ) c t b = 5 8 ℓ , 3/5 of which is b = v t b = 3 8 ℓ .