# Time dilation calculation

I saw this example in a book, where it portrays the following situation:

A guy is standing in a train travelling at a speed V and the train cart has a height of h, and there is a bulb attached to the top of the cart. The individual in the cart sees that the light reaches the ground in $$t_1 = h/c$$ an individual seeing the light from outside the cart sees the light reaching the floor in time: $$t_2 = \frac{\sqrt{h^2+(vt_1)^2}}{c}$$ If you simplify and replace $$h/c=t_1$$ in the second equation, you get the following: $$t_2=t_1\sqrt{1+\frac{v^2}{c^2}}$$ How do you get to the Lorentz factor from this equation?

• Do you know what the Lorentz factor is? Feb 5, 2021 at 6:59

Traveler-guy says the light hits the ground at time $$t_1$$. This is reflected in your first equation.
To get your second equation, you presumably drew a triangle with height $$h$$ and length $$vt_1$$. This means you implicitly assumed that according to station-guy, the light travels a horizontal distance $$vt_1$$ before hitting the ground. But that's wrong --- according to station-guy, the train is traveling at velocity $$v$$ for $$t_2$$ seconds, not $$t_1$$ seconds, before the light hits. So your triangle should have had a base of $$vt_2$$, not $$vt_1$$.