Derive the differential equation(s) describing the given physical situation. By Newton&...

Derive the differential equation(s) describing the given physical situation.

By Newton's universal law of gravitation the free-fall acceleration a of a body, such as the satellite shown in Figure 1.1.6, falling a great distance to the surface is not the constant g. Rather, the acceleration a is inversely proportional to the square of the distance r from the center of the earth, where k is the constant of proportionality.

(a) Use the fact that at the surface of the earth r = R and a = g to deter-mine the constant of proportionality k.

(b) Use Newton's second law and part (a) to find a differential equation for the distance r.

(c) Use the chain rule in the form

to express the differential equation in part (b) as a differential equation involving v and dv/dr.