Question

How fast does a rocket ship have to go for its length to be contracted to 99% of its rest length?

Answer 1

Answer:

4.2×10⁷ m/s

Explanation:

L = length observed by an observer

L₀ = Proper length

v = Velocity of the rocket ship

Length contraction formula

$L=L_{0}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\\\Rightarrow 0.99=1{\sqrt{1-\frac{v^{2}}{c^{2}}}}\\\Rightarrow \frac{0.99}{1}={\sqrt{1-\frac{v^{2}}{c^{2}}}}\\\Rightarrow 0.99^2=1-\frac{v^{2}}{c^{2}}\\\Rightarrow 0.99^2-1=-\frac{v^2}{c^2}\\\Rightarrow 0.0199\times c^2=v^2\\\Rightarrow v=\sqrt{0.0199}c=v\\\Rightarrow v =\sqrt{0.0199}\times 3\times 10^8\\\Rightarrow v=4.2\times 10^7\ m/s$

The speed of the ship would be 4.2×10⁷ m/s