Consider launching a satellite into orbit using a single-stage rocket. The rocket is continuously losing mass, which is being propelled away from it at significant speeds. We are interested in predicting the maximum speed the rocket can attain.²

a. Assume the rocket of mass m is moving with speed v. In a small increment of time Δt it loses a small mass Δm_p, which leaves the rocket with speed u in a direction opposite to v. Here, Δm_p is the small propellant mass. The resulting speed of the rocket is v + Δv. Neglect all external forces (gravity, atmospheric drag, etc.) and assume Newton's second law of motion:

b. Assume that initially, at time t = 0, the velocity v = 0 and the mass of the rocket is m = M + P, where P is the mass of the payload satellite and M = ∈M +(1–∈) M (0 M plus the mass (1–∈) M of the rocket casings and instruments. Solve the model in part (a) to obtain the speed:

c. Show that when all the fuel is burned, the speed of the rocket is given by:

d. Find v_f if c = 3 km/sec,  ∈ = 0.8, and β = 1/100. (These are typical values in satellite launchings.)

e. Suppose scientists plan to launch a satellite in circular orbit h km above the earth's surface. Assume that the gravitational pull toward the center of the earth is given by Newton's inverse square law of attraction:

where r is the universal gravitational constant,mis the mass of the satellite,M_e is the earth's mass, and R_e is the radius of the earth. Assume that this force must be balanced by the centrifugal force mv²/(h + R_e), where v is the speed of the satellite. What speed must be attained by a rocket to launch a satellite into an orbit 100 km above the earth's surface? From your computation in part (d), can a single-stage rocket launch a satellite into an orbit of that height?