A spaceship enters the solar system moving toward the Sun at a constant speed re

Question

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. How fast is the spaceship traveling towards the Sun? The radius of the orbit of Jupiter is 778 × 109 m, and that of the orbit of Mars is 228 × 109 m. 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. How fast is the spaceship traveling towards the Sun? The radius of the orbit of Jupiter is 778 × 109 m, and that of the orbit of Mars is 228 × 109 m.

0.438c

0.671c

0.522c

0.982c

0.319c

Explanation / Answer

Using time dilation,
t = to / sqrt(1 - v^2/c^2)
L = Lo * sqrt(1 – v^2/c^2)

Speed of spaceship is given by,
v = L/to
v = Lo * sqrt(1 – v^2/c^2) / (to)
v = Lo/to * sqrt( 1 - v^2/c^2)


to = 50.0 min = 50.0*60 = 3000 s
Lo = ( 778 – 228 ) × 10 m
Lo = 550 * 10^9 m

v = (550 * 10^9)/3000 *  sqrt( 1 - v^2/c^2)
v = 0.522 c

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