The length of a spaceship is measured to be exactly one-third of its proper length. What is the speed of the spaceship relative to the observer? (Answer: 0.9428 c)



Given:

Let

Proper Length = L₀

Measured Length = L = `\frac {1}{3}`L₀

Speed of light = c 


To Find:

Speed of the Space Ship relative to the observer = v =  ?  c



Solution:


The Length Contraction formula in special theory of relativity is given by: 


L = L₀ `\sqrt {1 - frac {v^2}{c^2}}`

`\frac {1}{3}`L₀ = L₀ `\sqrt {1 - frac {v^2}{c^2}}`

`\frac {1}{3}` =  `\sqrt {1 - frac {v^2}{c^2}}`

taking square both sides

`\(frac {1}{3})^2` =  `\(sqrt {1 - frac {v^2}{c^2}})^2`

`\frac {1}{9}` =  `\1 - frac {v^2}{c^2}`

`\frac {v^2}{c^2}` = 1 - `\frac {1}{9}`

`\frac {v^2}{c^2}` = `\frac {8}{9}`

v² =  `\frac {8}{9}` c²

Taking Square root on both sides

`\sqrt {v²}` = `\sqrt frac {8}{9}` `\sqrt {c²}`

or

v = 0.9428 c -----------------Ans.

Thus, if the speed of the spaceship becomes equal to 0.9428 times the velocity of the light c (v = 0.9428 c) then the length of space ship observed by the observer contracted to one-third of its proper length.



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