Lydia proposes to send an interstellar probe off in stages. The first stage, traveling radially outward at 0.500c relative to Earth, will send off a second stage traveling in the same direction as the first stage at a speed of 0.500c relative to the first stage. Finally, the second stage will send off a third stage traveling in the same direction at 0.500c relative to the second stage. Relative to Earth, the speed of the third stage will have a magnitude of
A)0.500c.
B)0.800c.
C)0.929c.
D)0.972c.
E)1.500c.

Question

Quizplus · Accepted Answer

We can solve this problem using the relativistic velocity addition formula:
v = (u + w)/(1 + uw/c^2)
where v is the relative velocity between the first stage and Earth, u is the velocity of the first stage relative to Earth, w is the velocity of the second stage relative to the first stage, and c is the speed of light.

For the first stage, u = 0.500c and w = 0 (since it is not moving relative to itself). Plugging these values into the velocity addition formula gives:
v = (0.500c + 0)/(1 + 0*0/c^2) = 0.500c

For the second stage, u = 0.500c (relative to the first stage) and w = 0.500c (since it is moving in the same direction as the first stage). Plugging these values into the velocity addition formula gives:
v = (0.500c + 0.500c)/(1 + 0.500c*0.500c/c^2) = 0.800c

For the third stage, u = 0.500c (relative to the second stage) and w = 0.500c (since it is also moving in the same direction as the first two stages). Plugging these values into the velocity addition formula gives:
v = (0.500c + 0.500c)/(1 + 0.500c*0.500c/c^2) = 0.866c

However, we are asked for the speed of the third stage relative to Earth. To find this, we need to add the velocities of the first two stages using the velocity addition formula again. The velocity of the second stage relative to Earth is already known to be 0.500c, so we can use this as the value of u for the third stage. Plugging in u = 0.500c and w = 0.866c (the velocity we just calculated for the third stage relative to the second stage), we get:
v = (0.500c + 0.866c)/(1 + 0.500c*0.866c/c^2) = 0.929c

Therefore, the best choice is C, 0.929c.

Lydia Proposes to Send an Interstellar Probe Off in Stages