Page 466 - Introduction to Continuum Mechanics

450 Integral Formulation of General Principles

In particular, if the control volume has only translation (acceleration = a 0)with respect to the
inertial frame and no rotations, then we have

Example 7.8.1
A rocket of initial total mass M 0 moves upward while ejecting a jet of gases at the rate of
y unit of mass per unit time. The exhaust velocity of the jet relative to the rocket is v r and the
gage pressure in the jet of area A is p. Derive the differential equation governing the motion
of the rocket and find the velocity as a function of time. Neglect drag forces.

Fig. 7.7

Solution. Let V r be a control volume which moves upward with the rocket. Then relative to
K n the net x momentum outflux is —yv e. The motion of gases due to internal combustion does
not produce any net momentum change relative to the rocket, therefore, there is no rate of
change of momentum inside the control volume. The net surface force on the control volume
is an upward force of pA and the body force is (M 0-yt)g downward. However, since the
control volume is moving with the rocket which has an acceleration x , therefore, the term
x(M-yt) is to be added to the other force terms [See Eq. (7.8.2)]. Thus,