If an external force F acts upon a system whose mass varies with time, Newton’s law of motion isIn this equation, m is the mass of the system at time t, ν is its velocity, and ν + u is the velocity of the mass that is entering (or leaving) the system at the rate dm/dt. Suppose that a rocket of initial mass m₀ starts from rest, but is driven upward by firing some of its mass directly backward at the constant rate of dm/dt = -b units per second and at constant speed relative to the rocket u = -c. The only external force acting on the rocket is F = -mg due to gravity. Under these assumptions, show that the height of the rocket above the ground at the end of t seconds (t small compared to m₀/b) is

d(mv) dt F + (v + u) dm dt