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Chapter 36: Q. 16 (page 1059)

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

Expert verified

$a .) v = 0.9965 c b .) 鈭� x = 119.58 l y$

Step by step solution

01

Part (a) Step 1: Given Information

We have given that,

鈭� 蟿 = 10 years 鈭� t = 120 years

02

Part (a) Step 2: Simplify

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

$鈭� t = \frac{鈭� 蟿}{\sqrt{1 - 尾^{2}}} 鈭� t = \frac{1}{\sqrt{1 - 尾^{2}}} \frac{120}{10} = \frac{1}{\sqrt{1 - {(\frac{v}{c})}^{2}}} (Substitutevaluesinequation.) 12 = \frac{1}{\sqrt{1 - {(\frac{v}{c})}^{2}}}$

$1 - {(\frac{v}{c})}^{2} = {(\frac{1}{12})}^{2} {(\frac{v}{c})}^{2} = 1 - {(\frac{1}{12})}^{2} \frac{v}{c} = \sqrt{1 - {(\frac{1}{12})}^{2}} v = \sqrt{1 - {(\frac{1}{12})}^{2} c} v = 0.9965 c$

03

Part (b) Step 1: Given Information

We have given that,

鈭� 蟿 = 10 years 鈭� t = 120 years

04

Part (b) Step 2: Simplify

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

$鈭� x = v 鈭� t 鈭� x = 0.9965 c 脳 (120 y) (Substitutevaluesinequation.) 鈭� x = 0.9965 \frac{l y}{y} 脳 (120 y) 鈭� x = 119.58 l y$

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