Simple Nature - Light and Matter

to deduce the general equation for gravitational energy. The equalarealaw turns out to be a statement on conservation of angularmomentum, which is discussed in chapter 4. We’ll demonstratethe elliptical orbit law numerically in chapter 3, and analytically inchapter 4.2.3.2 Circular orbitse / A cannon fires cannonballsat different velocities, fromthe top of an imaginary mountainthat rises above the earth’s atmosphere.This is almost the sameas a figure Newton included in hisMathematical Principles.Kepler’s laws say that planets move along elliptical paths (withcircles as a special case), which would seem to contradict the proofon page 90 that objects moving under the influence of gravity haveparabolic trajectories. Kepler was right. The parabolic path wasreally only an approximation, based on the assumption that thegravitational field is constant, and that vertical lines are all parallel.In figure e, trajectory 1 is an ellipse, but it gets chopped off whenthe cannonball hits the earth, and the small piece of it that is aboveground is nearly indistinguishable from a parabola. Our goal isto connect the previous calculation of parabolic trajectories, y =(g/2v 2 )x 2 , with Kepler’s data for planets orbiting the sun in nearlycircular orbits. Let’s start by thinking in terms of an orbit thatcircles the earth, like orbit 2 in figure e. It’s more natural nowto choose a coordinate system with its origin at the center of theearth, so the parabolic approximation becomes y = r − (g/2v 2 )x 2 ,where r is the distance from the center of the earth. For smallvalues of x, i.e., when the cannonball hasn’t traveled very far fromthe muzzle of the gun, the parabola is still a good approximationto the actual circular orbit, defined by the Pythagorean theorem,r 2 = x 2 + y 2 , or y = r √ 1 − x 2 /r 2 . For small values of x, we can usethe approximation √ 1 + ɛ ≈ 1+ɛ/2 to find y ≈ r−(1/2r)x 2 . Settingthis equal to the equation of the parabola, we have g/2v 2 = (1/2r),orv = √ gr [condition for a circular orbit] .Low-earth orbit example 14To get a feel for what this all means, let’s calculate the velocityrequired for a satellite in a circular low-earth orbit. Real low-earthorbitsatellites are only a few hundred km up, so for purposes ofrough estimation we can take r to be the radius of the earth, andg is not much less than its value on the earth’s surface, 10 m/s 2 .Taking numerical data from Appendix 5, we havev = √ gr√= (10 m/s 2 )(6.4 × 10 3 km)√= (10 m/s 2 )(6.4 × 10 6 m)√= 6.4 × 10 7 m 2 /s 2= 8000 m/s(about twenty times the speed of sound).98 Chapter 2 Conservation of Energy