Theory and Applications

Height of a rocket If an external force F acts upon a system whose mass varies with time, Newton’s law of motion is

In this equation, m is the mass of the system at time t, y is its velocity, and v + 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