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Another Reason You Can't Go as Fast as Light

The preceding argument was not very rigourous, but it served to show the essential necessity for regarding the EFFECTIVE MASS of an object as a relative quantity. Let's see what happens as we try to accelerate a mass to the velocity of light: at first it picks up speed just as we have been trained to expect by Galileo.^24.2 But as $\beta$ becomes appreciable, we begin to see an interesting phenomenon: it gets harder to accelerate! (This is, after all, what we mean by ``effective mass.'') As $\beta \to 1$,  the multiplicative ``mass correction factor'' $\gamma \to \infty$  and eventually we can't get any more speed out of it, we just keep pumping energy into the effective mass. This immediately suggests a new way of looking at mass and energy, to be developed in the following section.

But first let's note an interesting side effect: the rate at which a constant accelerating force produces velocity changes, as measured from a nonmoving reference frame, slows down by a factor $1/\gamma$;  but the same factor governs the TIME DILATION of the ``speed'' of the clock in the moving frame. So (as observed from a stationary frame) the change in velocity per tick of the clock in the moving frame is constant. This has no practical consequences that I know of, but it is sort of cute.