A rocket traveling with constant velocity makes a 8.4 *10¹⁵ m trip in 1 year. The proper time in years between events which mark the beginning and end of the trip is: A) 0.21 B) 0.46 C) 1.0 D) 2.2 E) 4.7

Question

Quizplus · Accepted Answer

We can use the time dilation formula 
$$\Delta t = \frac{\Delta 	au}{\sqrt{1-\frac{v^2}{c^2}}}.$$ 
 
Since the rocket is traveling at a constant velocity, its speed is $v = \frac{d}{t} = \frac{8.4	imes 10^{10}	ext{ m}}{1	ext{ year}}.$ Plugging in numbers, we get:
 
\begin{align*} 
\Delta t &= \frac{\Delta 	au}{\sqrt{1-\frac{v^2}{c^2}}} \ 
\Delta 	au &= \Delta t \sqrt{1-\frac{v^2}{c^2}} \
&= 1	ext{ year}\cdot \sqrt{1-\frac{\left(\frac{8.4	imes 10^{10}	ext{ m}}{1	ext{ year}}ight)^2}{(3	imes 10^8	ext{ m/s})^2}} \
&\approx 0.46	ext{ years}. 
\end{align*} 
 
So the answer is $\boxed{	extbf{(B)}}.$

A Rocket Traveling with Constant Velocity Makes a 8