Simple Nature - Light and Matter

3.1 Momentum In One Dimensiona / Systems consisting of materialparticles that interact through anenergy U(r). Top: The galaxyM100. Here the “particles” arestars. Middle: The pool ballsdon’t interact until they come togetherand become compressed;the energy U(r) has a sharpupturn when the center-to-centerdistance r gets small enoughfor the balls to be in contact.Bottom: A uranium nucleusundergoing fission. The energyU(r) has a repulsive contributionfrom the electrical interactionsof the protons, plus an attractiveone due to the strong nuclearinteraction. (M100: Hubble SpaceTelescope image.)3.1.1 Mechanical momentumIn the martial arts movie Crouching Tiger, Hidden Dragon, thosewho had received mystical enlightenment are able to violate the lawsof physics. Some of the violations are obvious, such as their abilityto fly, but others are a little more subtle. The rebellious youngheroine/antiheroine Jen Yu gets into an argument while sitting at atable in a restaurant. A young tough, Iron Arm Lu, comes runningtoward her at full speed, and she puts up one arm and effortlesslymakes him bounce back, without even getting out of her seat orbracing herself against anything. She does all this between bites.Although kinetic energy doesn’t depend on the direction of motion,we’ve already seen on page 89 how conservation of energy combinedwith Galilean relativity allows us to make some predictionsabout the direction of motion. One of the examples was a demonstrationthat it isn’t possible for a hockey puck to spontaneously reverseits direction of motion. In the scene from the movie, however,the woman’s assailant isn’t just gliding through space. He’s interactingwith her, so the previous argument doesn’t apply here, andwe need to generalize it to more than one object. We consider thecase of a physical system composed of pointlike material particles,in which every particle interacts with every other particle throughan energy U(r) that depends only on the distance r between them.This still allows for a fairly general mechanical system, by which Imean roughly a system made of matter, not light. The charactersin the movie are made of protons, neutrons, and electrons, so theywould constitute such a system if the interactions among all theseparticles were of the form U(r). 1 We might even be able to getaway with thinking of each person as one big particle, if it’s a goodapproximation to say that every part of each person’s whole bodymoves in the same direction at the same speed.The basic insight can be extracted from the special case wherethere are only two particles interacting, and they only move in onedimension, as in the example shown in figure b. Conservation ofenergy saysK 1i + K 2i + U i = K 1f + K 2f + U f .For simplicity, let’s assume that the interactions start after the timewe’re calling initial, and end before the instant we choose as final.This is true in figure b, for example. Then U i = U f , and we cansubtract the interaction energies from both sides, giving.K 1i + K 2i = K 1f + K 2f12 m 1v1i 2 + 1 2 m 2v2i 2 = 1 2 m 1v1f 2 + 1 2 m 2v2f 2 .1 Electrical and magnetic interactions don’t quite behave like this, which is apoint we’ll take up later in the book.130 Chapter 3 Conservation of Momentum