Rocket Motion Suppose a small single-stage rocket of total mass m(t) is launched ver...

Rocket Motion Suppose a small single-stage rocket of total mass m(t) is launched vertically and that the rocket consumes its fuel at a constant rate. If the positive direction is upward and if we take air resistance to be linear, then a differential equation for its velocity v(t) is given by

where k is the drag coefficient, λ is the rate at which fuel is consumed, R is the thrust of the rocket, m0 is the total mass of the rocket at t = 0, and g is the acceleration due to gravity.(See Problem 29 in Chapter 1 in Review.)

(a) Find the velocity v(t) of the rocket if m₀ = 200 kg, R = 2000 N, λ = 1 kg/s, g = 9.8 m/s², k = 3 kg/s, and v(0) = 0.

(b) Use ds/dt = v and the result in part (b) to find the height s(t) of the rocket at time t.