A rocket is fired straight up, burning fuel at the constant rate of b kilograms per second. Let v = v ( t ) be the velocity of the rocket at time t and suppose that the velocity u of the exhaust gas is constant. Let M = M ( t ) be the mass of the rocket at time t and note that M decreases as the fuel burns. If we neglect air resistance, it follows from Newton’s Second Law that

F = M d v d t − u b

where the force F = − M g . Thus

1 M d v d t − u b = − M g

Let M i be the mass of the rocket without fuel, M 2 the initial mass of the fuel, and M 0 = M 1 + M 2 . Then, until the fuel runs out at time t = M 2 / b , the mass is M = M 0 − b t .

(a) Substitute M = M 0 − b t 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 = M 2 / 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.