At what speed would an astronaut need to travel so that he would age 4.20 years while traveling a distance of 4.20 ly as measured by a stationary observer?
A) 1.00 c
B) This cannot be done.
C) 1.00 c/2
D) .707 c
E) .750 c

Question

Quizplus · Accepted Answer

This problem can be solved using the concept of time dilation. From the perspective of the stationary observer on Earth, the astronaut's time is dilated or slowed down due to their high velocity. The equation for time dilation is given by:

t' = t / sqrt(1 - v^2/c^2)

Where t' is the dilated time, t is the proper time (time for the stationary observer), v is the velocity of the astronaut, and c is the speed of light.

We can solve for the velocity v by plugging in the given values:

4.20 years = t' = t / sqrt(1 - v^2/c^2)
4.20 ly = ct' = ct / sqrt(1 - v^2/c^2)
4.20 = t / sqrt(1 - v^2/c^2)

Squaring both sides of the last equation and solving for v^2/c^2 gives:

v^2/c^2 = 1 - (t'/t)^2 = 1 - (4.20/4.20)^2 = 0.25

Taking the square root of both sides gives:

v/c = 0.5

So the velocity of the astronaut relative to the stationary observer is 0.5 times the speed of light, or:

v = 0.5c

Therefore, the correct answer is choice D, 0.707 c.

At What Speed Would an Astronaut Need to Travel So