Relativity Lite

Gravity Lite | 57 What happens if I were to throw a pen straight up? (It goes up, stops, and then falls back down.) Suppose I wanted it to go higher. What would I do? (Throw it faster.) Suppose I wanted it to travel to an infinite height before it stops and falls down again. How fast would I need to throw it? Pretend we are on an airless planet. The easy way to find this is to use energy conservation. Actually, let us run the movie backward. Suppose we have a pen sitting at infinity and let it fall toward the Earth. How fast will it be moving just as it hits the surface? Let me just quote the result: the speed with which a pen falling from infinity would hit the Earth is v GM R E E = = × × × × × − 2 2 6 67 10 5 10 6 378 11 24 . . m /s /kg .95 kg 3 2 10 1 24 10 11 2 6 8 2 m m 2 v s km s E = × = . / . / * * The derivation is not conceptually difficult, but it is long so I put it in this footnote for the curious. Feel free to ignore it. We introduced a formal definition of the potential energy of the gravitational field when we talked about a marble rolling in a bowl (figure 6). You can see that that marble gains kinetic energy ( E mv K = 1 2 2 from chapter 2) as it moves toward the bottom of the bowl and then loses it again as it climbs the far wall. Likewise, if we look at comets in highly elliptical orbits about the Sun, they move very fast near the Sun and slow down as they move away. They give up the potential energy that they have far from the Sun, and it is converted into kinetic energy (higher speeds) near perihelion, and that kinetic energy is converted back to potential energy as they return to the outer edges of the Solar System. The potential energy of the Earth’s gravitational field at the surface is given by E GMm r P = − . Why the minus sign? Well, it is convenient to define the total energy, E E mv GMm r K P + = − 1 2 2 , as being zero at infinity (where the pen is at rest for a moment), E m GMm i = − ∞ = − = 1 2 0 0 0 0 2 . Using energy conservation in this way we find that the final energy of the pen after falling from infinity will also sum to zero: E mv GMm R f E E = − = 1 2 0 2 . Adding the same thing to both sides of an equality will not change that equality, so 1 2 0 2 mv GMm R GMm R GMm R E E E E − + = + , or 1 2 2 mv GMm R E E = . Multiplying or dividing both sides of an equality by the same thing will not change that equality, so 2 1 2 2 2 m mv m GMm R E E = , or v GM R E E 2 2 = . Taking the square root of both sides of an equality will not change that equality, so v GM R E E = = × × × × × − 2 2 6 67 10 5 10 6 378 11 24 . . m /s /kg .95 kg 3 2 10 6 m .