A spaceship enters the solar system moving toward the Sun at a constant speed relative to the Sun. By its own clock, the time elapsed between the time it crosses the orbit of Jupiter and the time it crosses the orbit of Mars is 50.0 minutes. As measured in the coordinate system of the Sun, how long did it take for the spaceship to travel that distance? The radius of the orbit of Jupiter is 778 × 10⁹ m, and that of the orbit of Mars is 228 × 10⁹ m. A) 30.0 minutes B) 43.6 minutes C) 58.6 minutes D) 69.8 minutes E) 95.8 minutes

Question

Quizplus · Accepted Answer

We can solve this problem using the time dilation formula:
Δt' = Δt / √(1 - v^2/c^2)
where Δt is the time elapsed according to the spaceship's clock, Δt' is the time elapsed according to the Sun's coordinate system, v is the spaceship's speed relative to the Sun, and c is the speed of light.

We are given Δt = 50.0 minutes and we can find v using the distance between the orbits of Jupiter and Mars:
d = 778 × 10^6 m + 228 × 10^6 m = 1.006 × 10^9 m
Δt = d/v
v = d/Δt = (1.006 × 10^9 m)/(50.0 × 60.0 s) = 3.35 × 10^4 m/s

Plugging in the values into the formula, we get:
Δt' = 50.0 min / √(1 - (3.35 × 10^4 m/s)^2/(299792458 m/s)^2) ≈ 58.6 minutes

Therefore, the answer is choice C, 58.6 minutes.

A Spaceship Enters the Solar System Moving Toward the Sun