transformation for velocities in the y-dir or for any angle; these can be orked out as needed. Inside velocity is vr, which means that the displacement x is equal to the velocity times the time Now we have only to calculate what the position and time are from the point of view of the outside observer for an object which has the relation(16.2)between x'and r,. So we simply substitute (16.)into(16. 2), and obtain x=√1-v2/C2 (164) But here we find x expressed in terms of r. In order to get the velocity as seen by side, we must divide his distance by his time, not by the other man's time! So we must also calculate the time as seen from the outside, which is (16.5) Now we must find the ratio of x to t, which is +Ur (166) the square roots having cancelled. This is the law that we seek: the resultant ve- locity, the"summing "of two velocities, is not just the algebraic sum of two veloc- ities(we know that it cannot be or we get in trouble), but is"corrected"by Now let us see what happens. Suppose that you are moving inside the space ship at half the speed of light, and that the space ship itself is going at half the speed light. Thus u is yc and v is 2c, but in the denominator uv is one-fourth, so that So, in relativity,"half"and“half" does not make“one," it makes only“4/5.” Of course low velocities can be added quite easily in the familiar way, because so long as the velocities are small compared with the speed of light we can forget about the(1 uu/c2)factor; but things are quite diferent and quite interesting at high velocity. et us take a limiting case. Just for fun, suppose that inside the space ship the man was observing light itself. In other words, v c, and yet the space ship is moving. How will it look to the man on the ground? The answer will be 1+ uc/c Therefore, if ething is moving at the speed of light inside the ship. to be moving at the speed of light from the point of view of the man too! This is good, for it is, in fact, what the Einstein theory of designed to do in the first place-so it had better work Of course. there are cases in which the motion is not in the direction of the uniform translation. For example, there may be an object inside the ship which is just moving"upward"with the velocity v,, with respect to the ship, and the ship is moving"horizontally "Now, we simply go through the same thing, only using ys instead of x's, with the result so that if vr,= 0 yy= y= wyr, (167)