Suppose a small single-stage rocket of total mass m(t) is launched vertically, the positive direction is upward, the air resistance is linear, and the rocket consumes its fuel at a constant rate. In Problem 22 of Exercises 1.3 you were asked to use Newton’s second law of motion in the form given in (17) of that exercise set to show that a mathematical model for the velocity v(t) of the rocket is given by

dv/dt + {(k – λ) ­/(m₀ – ­λ_t)}v = - g + R/(m₀ – λ­t),

where k is the air resistance constant of proportionality, ­ is the constant rate at which fuel is consumed, R is the thrust of the rocket, m(t) = m₀ – λ_t, m₀ is the total mass of the rocket at t = 0, and g is the acceleration due to gravity.

(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 (a) to fi­nd the height s(t) of the rocket at time t.