Question

An observer on a spacecraft moving at 0.700c relative to the earth finds that a car takes 40.0 min to make a trip. How long does the trip take to the driver of the car.

Answer 1

✪ᴀɴsᴡᴇʀ✪

• The journey will take 28.57 min according to the driver.

★ᴇxᴘʟᴀɪɴᴀᴛɪᴏɴ★

Let

• Speed of spacecraft w. r. t car be 'v'.
• Time of journey as observed from spacecraft = t = 40 min
• Time of journey as observed by the driver(proper time) be $t_0$

Assume that speed of car is negligible(as it is not given) compared to speed of spacecraft. Thus, speed of the spacecraft w. r. t the car is approximated as speed of spacecraft w. r. t. earth.

$\implies v ≈ 0.7c$

Then the time of journey as observed from the spacecraft is given by

$\to t = \frac{t_0} {\sqrt{1-\frac{v^2}{c^2}}}$

$\to 40\:min= \frac{t_0} {\sqrt{1-\frac{{(0.7c) }^{2}}{c^2}}}$

$\to 40\sqrt{1- {0.7}^{2}}= t_0$

$\to 40\sqrt{0.51}= t_0$

$\to t_0 = 28.57\:min$

Thus, according to driver, the journey takes 28.57 min.