In this problem we explore a slightly more realistic model of a rocket's acceleration still neglecting air resistance. Consider a rocket fired straight up from rest burning fuel at the constant rate of b kg/s. 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 is not constant. From Newton's second law it can be shown that F=m(dv/dt) - uv where the force F= -mg and g is acceleration due to gravity, thus, m(dv/dt) - uv = -mg. Let M1 be the mass of the rocket without fuel and M2 be the initial mass of the fuel.
a. Find an equation for the mass m at time t in terms of M1, M2, and b.
b. Substitute this expression for m in equation one and solve the resulting equation for b. Use separation of variables.
c. Determine the velocity of the rocket at the time that the fuel is exhausted. This is called the burnout velocity.
d. Find the height of the rocket at the burnout time.
See Newton's 2nd Law above.
The Attempt at a Solution
I honestly don't even know how to start this problem, so any help would be greatly appreciated. Thank you!