Why Cats Land on Their Feet by Levi Mark

Author:Levi, Mark
Language: eng
Format: epub
ISBN: 9780691148540
Publisher: Princeton University Press

Figure 8.7. Acceleration is in direct proportion to (velocity)2.

Here is the same problem, restated:

A rocket problem. A toy rocket flies in a circle of radius 1m, with its thrusters aimed at a fixed angle α = 45° to the trajectory (Figure 8.7).11 How long will it take the rocket to increase its speed from 1m/sec to the speed of sound? to the speed of light?

Solution. The rock(et) will exceed the speed of light—not to mention the speed of sound—in less than 1 second! In fact, the velocity approaches infinity as the time approaches the 1-second mark. This just means that it is impossible in principle to spin the rock in a circle, forever maintaining the constant lead angle of 45° (or any other nonzero angle). Here is an explanation.

An explanation. I will show that the rock’s tangential acceleration a is in direct proportion the square of the velocity υ—in fact, that a = υ2. That is, the rate of change of υ is directly proportional to υ2. And by a calculus argument of the next paragraph, such a quantity becomes infinite in finite time.

Precise details. Since the angle between the tension force T and the tangent is 45°, we conclude from Figure 8.7 that the tangential and the radial components of T are equal. The same is true then for the tangential and the centripetal accelerations: a = ac. But the centripetal acceleration (page 181) is given by ac = v2/r = v2 (recall that r = 1m), and we conclude

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