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 Iong did it take for the spaceship to travel that distance? The radius of the orbit of Jupiter is $778 \times$ $10 ^ { 9 } \mathrm {~m}$, and that of the orbit of Mars is $228 \times 10 ^ { 9 } \mathrm {~m}$.
A) $58.6$ minutes
B) $30.0$ minutes
C) $95.8$ minutes
D) $69.8$ minutes
E) $43.6$ minutes

Question

Quizplus · Accepted Answer

The time dilation formula, $\Delta t' = \Delta t / \sqrt{1 - v^2/c^2}$, where $\Delta t'$ is the time interval in the moving frame (50.0 minutes in this case), and $\Delta t$ is the time interval in the stationary frame (what we're solving for), shows that the time interval in the stationary frame (the sun's frame) will always be longer than in the moving frame due to the factor of $\sqrt{1 - v^2/c^2}$ being less than 1. Therefore, the only option longer than 50.0 minutes is 58.6 minutes.

A Spaceship Enters the Solar System Moving Toward the Sun 50.050.050.0

A Spaceship Enters the Solar System Moving Toward the Sun $50.0$