Question

According to an observer on Earth, a spacecraft whizzing by at 0.6c is 35 m long. What is the length of the spacecraft according to the passengers onboard?

Short answer

The length of the spacecraft according to the passenger onboard is 43.75 m.

Step-by-step solution

Length contraction

Length contraction is the phenomenon where a moving object’s length is measured to be shorter than its proper length.Hence, the length of spacecraft according to an observer on earth will be shorter than its original length according to the passengers on board.

Length contraction is given by

$l=l_0\sqrt{1-\frac{v^2}{c^2}}$

Given Data

Velocity of spacecraft$\left(V\right)=0.6cm/s$

Length of spacecraft according to an observer on earth$\left(l\right)=35m$

Speed of light(c) = $3\times {10}^8\text{m/s}$

We have to find,

Length of spacecraft according to observer onboard$\left(l_{\circ }\right)$

Calculations

When we use the formula from step 1 and rearrange it in terms of known values, we get the following,

$\begin{array}{l}l=l_{\circ }\sqrt{1-\frac{v^2}{c^2}}\\ l_{\circ }=\frac{l}{\sqrt{1-\frac{v^2}{c^2}}}\\ l_{\circ }=\frac{35m}{\sqrt{1-\frac{{\left(0.6c\right)}^2}{c^2}}}\\ l_{\circ }=\frac{35m}{\sqrt{1-{\left(0.6\right)}^2}}\end{array}$

Simplifying further,

$\begin{array}{l}{l}_{\circ }=\frac{35m}{\sqrt{0.64}}\\ l_{\circ }=\frac{35m}{0.8}\\ l_{\circ }=43.75m\end{array}$

Hence, the original length of the spacecraft is.$43.75\text{\hspace{0.17em}m}$