Question

Suppose you are planning a trip in which a spacecraft is to travel at a constant velocity for exactly six months, as measured by a clock on board the spacecraft, and then return home at the same speed. Upon return, the people on earth will have advanced exactly 120 years into the future. According to special relativity, how fast must you travel

Answer 1

I must travel with a speed of 2.97 x 10^8 m/s

Step-by-step explanation:

Sine the spacecraft flies at the same speed in the to and fro distance of the journey, then the time taken will be 6 months plus 6 months

Time that elapses on the spacecraft = 1 year

On earth the people have advanced 120 yrs

According to relativity, the time contraction on the spacecraft is gotten from

`t` =
`t_(0) /\sqrt{1 - \beta ^(2) }`

where

`t` is the time that elapses on the spacecraft = 120 years

`t_(0)` = time here on Earth = 1 year

`\beta` is the ratio v/c

where

v is the speed of the spacecraft = ?

c is the speed of light = 3 x 10^8 m/s

substituting values, we have

120 = 1/
`\sqrt{1 - \beta ^(2) }`

squaring both sides of the equation, we have

14400 = 1/
`(1 - \beta ^(2) )`

14400 - 14400
`\beta ^(2)` = 1

14400 - 1 = 14400
`\beta ^(2)`

14399 = 14400
`\beta ^(2)`

`\beta ^(2)` = 14399/14400 = 0.99

`\beta = √(0.99)` = 0.99

substitute β = v/c

v/c = 0.99

but c = 3 x 10^8 m/s

v = 0.99c = 0.99 x 3 x 10^8 = 2.97 x 10^8 m/s