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Relative Velocity
 Consider two observers, A and B* moving towards each other as shown here. Suppose we are told that A moves at 3ms-1 and B moves at 2ms-1. Hearing this we will assume that, in one second, A moves 3m measured along the ground and similarly that B covers 2m in the same time. In other words we are assuming that these velocities are the velocities of the two observers relative to the ground. What is the velocity of B relative to A? To answer this question, imagine yourself to be A. What do you observe? In one second you move 3m forwards and B moves 2m in the opposite sense, so B moves 5m closer to you (that is, in the negative sense) in one second. So, if you could make a measurement** of the velocity of B, you would find 5ms-1 in the negative sense or -5ms-1 Also, if we imagine ourselves to be B, using the same logic we conclude that the velocity of A relative to B is +5ms-1. Now consider the following case. Suppose that A and B are both moving at 3ms-1 Imagining yourself to be A, you will conclude that you will never catch up with B. In other words, the velocity of B relative to A is zero. By imagining yourself to be one of the observers you are giving yourself zero velocity relative to that observer. In other words, mathematically, you are removing (subtracting) the velocity of that observer from your measurement. So, mathematically vB relative to A = vB relative to the ground - vA relative to the ground and, of course, there is nothing special about "the ground", it could be relative to any other body/reference point and, to save a bit of writing, I will use the following notation: vB(A) = vB(C) - vA(C) where C represents a third body (the ground in this case). Using the figures for the first situation above we have vB(A) = -2 - (+3) = -5ms-1 If all this seems pretty trivial, be warned... Einstein found that it is not quite true! However, the errors don't become a problem until the relative velocities involved are very large. More of that on other pages. One of Einstein's aims in developing the theory of relativity was to be able to take measurements made by one observer and transform them into measurements made by another observer. In other words, if one observer says that a body is moving at vms-1, what will another observer say about the same body? The answer to this question obviously depends on the relative motion of the two observers. We now introduce a third body***, p (for pelican, as you can see). Suppose that we do not know how fast A and B are moving relative to the ground but we do know that their relative velocity has magnitude u = 4ms-1 Suppose also that observer A has measured the velocity of p (relative to A). Let this velocity be v. If A tells B that p moves at 10ms-1 relative to A then B (observing that he is moving in the same sense as p) will conclude that p is moving at 6ms-1 in the negative sense, relative to him. Let this be v'. Clearly, this result is again obtained by a simple subtraction: vp(B) = vp(A) - vB(A) or v' = v - u v' = -10 -(-4) = -6 * I was going to use the letters A and G for obvious reasons: A for Albert and G for... ok, who knows? Yes, the great Groucho but eventually settled on boring old A and B. **How might this measurement be made in practice? See Measuring Relative Velocity *** Of course, we already had a "third body" before; the ground.
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