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

From the spaceship's point of view, the distance it travels is simply the difference in the radii of the orbits of Jupiter and Mars:

distance = (228 × 10^6 m) - (778 × 10^6 m) = -550 × 10^6 m

(Note that the negative sign just indicates that the spaceship is moving in the opposite direction.)

Using the time dilation formula, we can relate the time elapsed on the spaceship to the time elapsed in the coordinate system of the sun:

Δt_sun = Δt_ship / √(1 - v^2/c^2)

where v is the speed of the spaceship and c is the speed of light. Since the spaceship is moving at a constant speed relative to the sun, we can find v by dividing the distance traveled by the time elapsed on the spaceship:

v = -550 × 10^6 m / (50.0 min × 60s/min) = -183.3 km/s

(Note that the negative sign here just indicates that the spaceship is moving in the opposite direction.)

Plugging in the numbers:

Δt_sun = 50.0 min / √(1 - (-183.3 km/s)^2/c^2)

Δt_sun = 58.6 min

So the answer is (C) 58.6 minutes.

A Spaceship Enters the Solar System Moving Toward the Sun