Rocket Motion—Continued In Problem 44, suppose that of the rocket’s initial mass, 50...

Rocket Motion—Continued In Problem 44, suppose that of the rocket’s initial mass, 50 kg is the mass of the fuel.

(a) What is the burnout time t_b, or the time at which all the fuel is consumed? (See Problem 30 in Chapter 1 in Review.)

(b) What is the velocity of the rocket at burnout?

(c) What is the height of the rocket at burnout?

(d) Why would you expect the rocket to attain an altitude higher than the number in part (b)?

(e) After burnout what is a mathematical model for the velocity of the rocket?

Reference:

Problem 44:

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, l 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.