Question

On a futuristic highway, a $15m$long rocket travels so fast that a red stoplight, with a wavelength of $700nm$, appears to the pilot to be a green light with a wavelength of $520nm$. What is the length of the rocket to an observer standing at the intersection as the rocket speeds through?

Short answer

$L\text{'}=1.449\times {10}^{-2}km\to 14.49m$length of the rocket to an observer standing at the intersection as the rocket speeds through.

Step-by-step solution

Given Information

We need to find the length of the rocket to an observer standing at the intersection as the rocket speeds through.

Simplify 

Consider,

$L=15m\to 1.5\times {10}^{-2}km$

${\lambda }_0=700nm$

$\lambda =520nm$

First, finding $∆\lambda$,

$∆\lambda =\lambda -{\lambda }_0=520nm-700nm=-180nm$

Now finding velocity $v$:

$\frac{∆\lambda }{{\lambda }_0}=\frac{v}{c}$

$\Rightarrow v=c\frac{∆\lambda }{{\lambda }_0}$

$v=3\times {10}^{5}km/s\left(-\frac{180}{700}\right)$

$v=-7.714\times {10}^{4}km/s$

Finally, we will find the length of the rocket$L\text{'}$

$$\begin{gathered} L\text{'}=\sqrt[L]{1-\frac{v^2}{c^2}} \\ L\text{'}=1.5\times {10}^{-2}\sqrt{1-\frac{{\left(-7.714\times {10}^2\right)}^2}{{\left(3\times {10}^5\right)}^2}} \\ L\text{'}=1.449\times {10}^{-2}km\to 14.49m \end{gathered}$$