Question

A rocket ship is 100m in length when measured before leaving the launching pad. when in flight a ground observer measure its length as 75m. what is the velocity of rocket ship?

Answer 1

Using Relativity Length Formula

7/16 * speed of light ^2

image attached above

Answer 2

Let the original velocity of rocket  be  L and  observed L'

L= 100 m

L'  = 75 m

Using relative  length formula,

$\frac{L}{L'}=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\\\;\\\text{Where,c=velocity of light,\;\;v=velocity of rocket}\\\;\\\frac{100}{25}=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\\\;\\\frac{4}{3}=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\\\;\\\frac{3}{4}=\sqrt{1-\frac{v^{2}}{c^{2}}}\\\;\\\frac{9}{16}=1-\frac{v^{2}}{c^{2}}\\\;\\\frac{v^{2}}{c^{2}}=1-\frac{9}{16}\\\;\\\frac{v^{2}}{c^{2}}=\frac{16-9}{16}\\\;\\\frac{v^{2}}{c^{2}}=\frac{7}{16}\\\;\\v^{2}=\frac{7c^{2}}{16}\\\;\\v=\frac{\sqrt{7}c}{4}\\\;$

Putting  c = 3 × 10⁸ m/s

$v=\frac{\sqrt{7}\times3\times10^{8}}{4}\\\;\\v=1.98\times10^{8}\:m/s$