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## Homework Statement

A train moves at speed v. Bullets are successively fired at speed u (relative to the train) from the back of the train to the front. A new bullet is fired at the instant (as measured in the train frame) the previous bullet hits the front. In the frame of the ground, what fraction of the way along the train is a given bullet, at the instant (as measured in the ground frame) the next bullet is fired? What is the maximum number of bullets that are in flight at a given instant, in the ground frame?

## Homework Equations

Δt = γ(Δt' + vΔx'/c^2)

u = (u'+v)/(1+u'v/c^2)

## The Attempt at a Solution

In the ground frame, the time elapsed between the bullet being fired and hitting the front is given by the Lorentz transformation:

Δt = γ(Δt' + vΔx'/c^2)

If the train has proper length l', then in the train frame Δt' = l'/u' and Δx'=l', so this gives:

Δt = γ(l'/u' + vl'/c^2)

So in the ground frame, the next bullet is fired at Δt. The distance travelled by a bullet in this time in the ground frame is therefore uΔt.

Using u = (u'+v)/(1+u'v/c^2) and Δt = γ(l'/u' + vl'/c^2)

Distance = (u'+v)/(1+u'v/c^2)*γ(l'/u' + vl'/c^2)

Now when I multiply this out I just get a huge mess that doesn't simplify into anything useful, so I feel like I'm doing this completely wrong. Could anyone give me any guidance?

Thanks