A rocket is fired straight up, burning fuel at the constant rate of kilograms per second. Let v = v(t) be the velocity of the rocket at time and suppose that the velocity of the exhaust gas is constant. Let M = M(t) be the mass of the rocket at time and note that M decreases as the fuel burns. If we neglect air resistance, it follows from Newton’s Second Law that F = M dv/dt – ub where the force F = – Mg. Thus
Let M1 be the mass of the rocket without fuel, M2 the initial mass of the fuel, and M0 = M1 + m2 . Then, until the fuel runs out at time t = M2b, the mass is M = M0 - bt.
(a) Substitute M = M0 – bt into Equation 1 and solve the resulting equation for v. Use the initial condition v(0) = 0 to evaluate the constant.
(b) Determine the velocity of the rocket at time t = M2/b. This is called the burnout velocity.
(c) Determine the height of the rocket y = y(t) at the burnout time.
(d) Find the height of the rocket at any time t.