A)Suppose the speed of light were 100 mph. Trains on parallel tracks are approac

Question

A)Suppose the speed of light were 100 mph. Trains on parallel tracks are approaching each other at speeds of 91 . 11 mph and 85 . 7 mph. A person on one of the trains sees the two trains approaching each other at what speed? B)After the trains pass a crazy person on the 91.11 mph train fires a rifle with muzzle velocity 91.11 mph back at the other train. With what speed does the person on that train see the bullet approach? (A negative answer means the bullet is receding). Answer in units of mph

Explanation / Answer

Suppose that Alice measures an object moving at speed u. Suppose that Bob is moving at speed v relative to Alice -- v is what Alice measures Bob's velocity to be.

In Alice's coordinates, the object describes the path:

x = u t.

When we do the Lorentz transform to Bob's coordinates x', t', we find:

x' = (x vt)
t' = (t vx/c²)

The exact value of doesn't matter for this problem, but it is = 1/(1 (v/c)²). The reason that it doesn't matter is that Bob sees this object going at a speed u' = x' / t', so the terms cancel out:

u' = x' / t' = (x vt) / (t vx/c²)

Substituting x = u t and canceling out t gives:

u' = (u v) / (1 uv/c²)

Let us now say that Alice is on the ground. She sees Bob going at speed v = +0.857c (or you may choose the other train). She sees the other train coming in at u = -0.9111c.

Bob sees it coming in at:

u' = - (0.9111 + 0.857) c / (1 + 0.9111*0.857) = -0.9928 c.

This would be an oncoming speed of 99.28 mph if c were 100mph, which it is not.

Now imagine that Carol is the crazy person firing the gun. Suppose for the sake of sanity that she uses the same directions of coordinates that Alice did -- thus Bob is moving at v=+0.9928 c in her reference frame, and she thinks that she fired a bullet at speed +0.9111 c. after him.Then Bob measures its speed as:

u' = - (0.9111 0.9928) c / (1 0.9111*0.9928) = -0.8558 c.

There is a good reason for getting this number back. Carol thinks that Alice is traveling at speed 0.9111 c, and she also thinks that her bullet is traveling at speed 0.9111c in the same direction -- thus, she fired a bullet at speed 0 relative to Alice. So Alice can stick out her hand and catch the bullet as it falls down. Bob sees this and figures that the bullet is travelling at the same speed as Alice, which is -0.8558 c in his reference frame.

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