Simple Nature - Light and Matter

3.1.6 The center of mass frame of referencer / The same collision of twopools balls, but now seen inthe center of mass frame ofreference.s / The sun’s frame of reference.t / The c.m. frame.A particularly useful frame of reference in many cases is theframe that moves along with the center of mass, called the centerof mass (c.m.) frame. In this frame, the total momentum is zero.The following examples show how the center of mass frame can bea powerful tool for simplifying our understanding of collisions.A collision of pool balls viewed in the c.m. frame example 17If you move your head so that your eye is always above the pointhalfway in between the two pool balls, as in figure r, you are viewingthings in the center of mass frame. In this frame, the ballscome toward the center of mass at equal speeds. By symmetry,they must therefore recoil at equal speeds along the lines onwhich they entered. Since the balls have essentially swappedpaths in the center of mass frame, the same must also be true inany other frame. This is the same result that required laboriousalgebra to prove previously without the concept of the center ofmass frame.The slingshot effect example 18It is a counterintuitive fact that a spacecraft can pick up speedby swinging around a planet, if it arrives in the opposite directioncompared to the planet’s motion. Although there is no physicalcontact, we treat the encounter as a one-dimensional collision,and analyze it in the center of mass frame. Since Jupiter is somuch more massive than the spacecraft, the center of mass isessentially fixed at Jupiter’s center, and Jupiter has zero velocityin the center of mass frame, as shown in figure 3.1.6. The c.m.frame is moving to the left compared to the sun-fixed frame usedin figure 3.1.6, so the spacecraft’s initial velocity is greater in thisframe than in the sun’s frame.Things are simpler in the center of mass frame, because it is moresymmetric. In the sun-fixed frame, the incoming leg of the encounteris rapid, because the two bodies are rushing toward eachother, while their separation on the outbound leg is more gradual,because Jupiter is trying to catch up. In the c.m. frame,Jupiter is sitting still, and there is perfect symmetry between theincoming and outgoing legs, so by symmetry we have v 1f = −v 1i .Going back to the sun-fixed frame, the spacecraft’s final velocityis increased by the frames’ motion relative to each other. In thesun-fixed frame, the spacecraft’s velocity has increased greatly.Einstein’s motorcycle example 19We’ve assumed we were dealing with a system of material objects,for which the equation p = mv was true. What if our systemcontains only light rays, or a mixture of light and matter? As a collegestudent, Einstein kept worrying about was what a beam oflight would look like if you could ride alongside it on a motorcycle.In other words, he imagined putting himself in the light beam’s144 Chapter 3 Conservation of Momentum