# Bullet from train scenario, but with light instead

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1. Mar 18, 2015

### Fred Choi

I am a student, and a few days ago in physics class we were discussing about the 'bullet fired from a train scenario', where e.g. if a train is heading towards a certain direction at 200 km/h and a bullet is fired at 800 km/h, the bullet would travel at 1000 km/h relative to the ground. However, one my friends asked an interesting question that even my teacher struggled to answer:
What if a torch was shone in the same direction as the train was travelling? Would the wave travel at the speed of light plus the speed of the train? But after all, nothing travels faster than the speed of light. Please help me with this! Thanks

2. Mar 18, 2015

### Orodruin

Staff Emeritus
The simple addition of velocities is only valid for velocities much smaller than the speed of light. 800 km/h and 1000 km/h are both much smaller than the speed of light and therefore it is a (very) good approximation to assume simple addition of velocities. When you deal with velocities close to the speed of light, you need to use relativistic addition of velocities, which is given by
$$v' = \frac{v+u}{1+\frac{uv}{c^2}}.$$

Suggested exercise: Try this out for v = 800 km/h and u = 200 km/h and see how much the result differs from 1000 km/h.

3. Mar 18, 2015

### Staff: Mentor

It doesn't move at exactly 1000 km/hr relative to the ground. If $u$ is the speed of the train and $v$ is the speed of the bullet, the correct formula is not $u+v$, it is $(u+v)/(1+uv/c^2)$ where $c$ is the speed of light. The difference is completely unnoticeable at the sorts of speeds you're taking about, but it is there. (Google for "relativistic velocity addition" for more information).

Try setting $v=c$ in the formula above, see what you get for the speed of the wavefront... It's is actually kinda fun to see how it comes out.

[edit: Curses! Beaten by Orodruin again! In need a faster internet connectioon! ]