Question
At what speed the volume of an object shrinks to half its rest value?
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Answer
Let the volume (initial) be V.
$\text{v}'=\frac{\text{v}}{2}$
So, $\frac{\text{v}}{2}=\text{v}\sqrt{1-\frac{\text{v}^2}{\text{c}^2}}$
$\Rightarrow\frac{\text{c}}{2}=\sqrt{\text{c}^2-\text{v}^2}\Rightarrow\frac{\text{c}^2}{4}=\text{c}^2-\text{v}^2$
$\Rightarrow\text{v}^2=\text{c}^2-\frac{\text{c}^2}{4}=\frac{3}{4}\text{c}^2\Rightarrow\text{v}=\frac{\sqrt{3}}{2}\text{c}$
$\text{v}'=\frac{\text{v}}{2}$
So, $\frac{\text{v}}{2}=\text{v}\sqrt{1-\frac{\text{v}^2}{\text{c}^2}}$
$\Rightarrow\frac{\text{c}}{2}=\sqrt{\text{c}^2-\text{v}^2}\Rightarrow\frac{\text{c}^2}{4}=\text{c}^2-\text{v}^2$
$\Rightarrow\text{v}^2=\text{c}^2-\frac{\text{c}^2}{4}=\frac{3}{4}\text{c}^2\Rightarrow\text{v}=\frac{\sqrt{3}}{2}\text{c}$
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