Beginning at time \(t=0\), astronauts in a landing module are descending toward the surface of an airless moon with a downward initial velocity \(-\left|v_{0}ight|\) and altitude \(y=h\) above the surface. The gravitational field \(g\) is essentially constant throughout this descent. An onboard retrorocket can provide a fixed downward exhaust velocity \(u\). The astronauts need to select a fixed exhaust rate \(\lambda=|d m / d t|\) in order to provide a soft landing with velocity \(v=0\) when they reach the surface at \(y=0\).

(a) Explain briefly why Newton's second law for the module during its descent has the form

(b) Find the velocity \(v\) of the module as a function of time, in terms of \(\left|v_{0}ight|, u, m_{0}, \lambda\), and \(g\).

(c) During the descent its velocity is \(v=d y / d t\), negative because it is downward. Find an expression for \(y(t)\) in terms of \(\left|v_{0}ight|, g, u, \lambda, m_{0}\), and \(h\).

Transcribed Image Text:

dv m(t) dt n = dm m(t)g dt