Textbook Problem

A rocket is fired straight up, burning fuel at the constant rate of b kilograms per second. Let $v=v\left(t\right)$ be the velocity of the rocket at time t and suppose that the velocity u of the exhaust gas is constant. Let $M=M\left(t\right)$ 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

$F=M\frac{dv}{dt}-ub$

where the force $F=-Mg$. Thus

1 $M\frac{dv}{dt}-ub=-Mg$

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 $M=M_0-bt$.

(a) Substitute $M=M_0-bt$ into Equation 1 and solve the resulting equation for v. Use the initial condition $v\left(0\right)=0$ 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 $y=y\left(t\right)$ at the burnout time.

(d) Find the height of the rocket at any time t.