Question

A box at rest has the shape of a cube 2.6 m on a side. This box is loaded onto the flat floor of a spaceship and the spaceship then flies past us with a horizontal speed of 0.80c. What is the volume of the box as we observe it

Answer 1

Answer:

The observed volume of the box is 3.796 cubic meters.

Explanation:

The observed length is determined by the formula for the Length Contraction:

$L = \frac{L_{o}}{\gamma}$

Where:

$L$ - Proper length, measured in meter.

$\gamma$ - Lorentz factor, dimensionless.

The Lorentz factor is represented by the following equation:

$\gamma = \frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}} }}$

If $v = 0.8\cdot c$, then:

$\gamma = \frac{1}{\sqrt{1-\frac{0.64\cdot c^{2}}{c^{2}} }}$

$\gamma = \frac{1}{\sqrt{1-0.64}}$

$\gamma = \frac{5}{3}$

Therefore, the observed length is:

$L = \frac{3}{5}\cdot L_{o}$

Given that $L_{o} = 2.6\,m$, the observed length is:

$L = \frac{3}{5}\cdot (2.6\,m)$

$L = 1.56\,m$

The observed volume of the box is:

$V = L^{3}$

$V = (1.56\,m)^{3}$

$V= 3.796\,m^{3}$

The observed volume of the box is 3.796 cubic meters.