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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?

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  • $\begingroup$ Do you know what the Lorentz factor is? $\endgroup$
    – Martin C.
    Feb 5, 2021 at 6:59

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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 $vt_1$.

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  • $\begingroup$ Ohh my bad... thanks for pointing that out! $\endgroup$
    – NeuroEng
    Feb 5, 2021 at 12:09

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