Question

a. At what speed, as a fraction of $c,$must a rocket travel on a journey to and from a distant star so that the astronauts age $10$years while the Mission Control workers on earth age $120$years?

b. As measured by Mission Control, how far away is the distant star?

Short answer

$$\begin{gathered} a.)v=0.9965c \\ b.)∆x=119.58ly \end{gathered}$$

Step-by-step solution

Part (a) Step 1: Given Information

We have given that,

$$\begin{gathered} ∆\tau =10\text{years} \\ ∆t=120\text{years} \end{gathered}$$

Part (a) Step 2: Simplify

We have to find speed $v,$as a fraction of $c,$must a rocket move on a journey.

$$\begin{gathered} ∆t=\frac{∆\tau }{\sqrt{1-{\beta }^2}} \\ ∆t=\frac{1}{\sqrt{1-{\beta }^2}} \\ \frac{120}{10}=\frac{1}{\sqrt{1-{\left({\displaystyle \frac{v}{c}}\right)}^2}}\text{(Substitutevaluesinequation.)} \\ 12=\frac{1}{\sqrt{1-{\left({\displaystyle \frac{v}{c}}\right)}^2}} \end{gathered}$$

$$\begin{gathered} 1-{\left(\frac{v}{c}\right)}^2={\left(\frac{1}{12}\right)}^2 \\ {\left(\frac{v}{c}\right)}^2=1-{\left(\frac{1}{12}\right)}^2 \\ \frac{v}{c}=\sqrt{1-{\left(\frac{1}{12}\right)}^2} \\ v=\sqrt{1-{\left(\frac{1}{12}\right)}^{2}c} \\ v=0.9965c \end{gathered}$$

Part (b) Step 1: Given Information

We have given that,

$$\begin{gathered} ∆\tau =10\text{years} \\ ∆t=120\text{years} \end{gathered}$$

Part (b) Step 2: Simplify

We need to figure out how far away the distant star is.

$$\begin{gathered} ∆x=v∆t \\ ∆x=0.9965c\times (120y)\text{(Substitutevaluesinequation.)} \\ ∆x=0.9965\frac{ly}{y}\times (120y) \\ ∆x=119.58ly \end{gathered}$$