Two spaceships, each 100 m long in its own rest frame, approach Earth from opposite directions, with equal speeds relative to Earth. As measured by a passenger in one of the spaceships, the other spaceship is 96 m long. What is the speed of the spaceships?
A) c/7
B) c/6
C) c/5
D) c/4
E) c/3

Question

Quizplus · Accepted Answer

The correct answer is A because the speed of the spaceships is c/7. This can be calculated using the Lorentz transformation, which gives the length of an object in motion as:$$L = L_0/\gamma$$where L is the length of the object in motion, L_0 is the length of the object at rest, and γ is the Lorentz factor, which is given by:$$\gamma = 1/\sqrt{1 - v^2/c^2}$$where v is the speed of the object and c is the speed of light.In this case, the length of the spaceship in its own rest frame is 100 m, and the length of the spaceship as measured by a passenger in the other spaceship is 96 m. So, we have:$$96 m = 100 m/\gamma$$Solving for γ, we get:$$\gamma = 100 m/96 m = 1.042$$Substituting this value of γ into the equation for the Lorentz factor, we get:$$1.042 = 1/\sqrt{1 - v^2/c^2}$$Solving for v, we get:$$v = c/\sqrt{1 - 1.042^2} = c/7$$

Two Spaceships, Each 100 M Long in Its Own Rest