Answer to Question #226426 in Optics for Muhammad Ali

Question

A ladder 7.00 m long leans against a wall inside a spaceship. From the point of view of a person on the ship, the base of the ladder is 2.50 m from the wall. The spaceship moves past the Earth with a speed of 0.94c in a direction parallel to the floor of the ship. What is the length of the ladder as seen by an observer on Earth?

Answer 1

Given:

$l=7.00\:\rm m$

$l_x=2.50\:\rm m$

We have

$$l_y=\sqrt{l^2-l_x^2}=\sqrt{7.00^2-2.50^2}=6.54\:\rm m$$

The Lorentz transformations give

$$l_x'=l_x\sqrt{1-v^2/c^2}=2.50\sqrt{1-0.94^2}=0.853\:\rm m$$ $$l_y'=l_y=6.54\:\rm m$$

Hence, the length of the ladder as seen by an observer on Earth

$$l'=\sqrt{l_x'^2+l_y'^2}=\sqrt{0.853^2+6.54^2}=6.59\:\rm m$$