Solved Numerical Problem No.1 ( Unit-18: Dawn of Modern Physics); HSSC-II (Class-12); Physics (New Book/KPK Text Board, Peshawar) With Verified Answers

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.

Similar Questions:

The time period of a pendulum is measured to be 3s in Inertial frame of the pendulum. What is the period when measured by an observer moving with a speed of 0.95c with respect to the pendulum? (Answer: 9.65)

A particle called the pion lives on the average only about 2.6 ｘ10⁻⁸ s when at rest in the laboratory. It then changes to another form. How long would such a particle live when shooting through the space at 0.95c?[Ans. 8.3 x 10⁻⁸ s)

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