Serway & Jewett — Linear Momentum and Collisions: Problem 34

A rocket has total mass \( M_i = 360 \text{ kg} \), including \( M_{\text{fuel}} = 330 \text{ kg} \) of fuel and oxidizer. In interstellar space, it starts from rest at the position \( x = 0 \), turns on its engine at time \( t = 0 \), and puts out exhaust with relative speed \( v_e = 1\,500 \text{ m/s} \) at the constant rate \( k = 2.50 \text{ kg/s} \). The fuel will last for a burn time of \( T_b = M_{\text{fuel}}/k = 330 \text{ kg}/(2.5 \text{ kg/s}) = 132 \text{ s} \). (a) Show that during the burn the velocity of the rocket as a function of time is given by \( v(t) = -v_e \ln(1 - kt/M_i) \). (b) Make a graph of the velocity of the rocket as a function of time for times running from 0 to 132 s. (c) Show that the acceleration of the rocket is \( a(t) = \frac{k v_e}{M_i - kt} \). (d) Graph the acceleration as a function of time. (e) Show that the position of the rocket is \( x(t) = v_e \left( \frac{M_i}{k} - t \right) \ln \left( 1 - \frac{kt}{M_i} \right) + v_e t \). (f) Graph the position during the burn as a function of time.