A Classical Approach of Newtonian Mechanics

6 CONSERVATION OF MOMENTUM 6.2 Two-component systems

vr vb

cannon cannonball

ground

Figure 47: Another example two-component system

equilibrium is not upset). Hence, there is zero net external force acting on the

system. It follows, from the previous discussion, that the centre of mass of the

system is subject to Newton’s first law. In particular, since the centre of mass is

clearly stationary before the winch is turned on, it must remain stationary both

during and after the period in which the winch is operated. Hence, the height of

the centre of mass,

is a conserved quantity.

xcm =^

mg xg + mw xw

, (6.11)

mg + mw

Suppose that the operation of the winch causes the height of the sandbag to

change by ∆xw, and that of the gondola to simultaneously change by ∆xg. If xcm
is a conserved quantity, then we must have

0 = mg ∆xg + mw ∆xw, (6.12)

or (^) m
w
∆xg = −
g
∆xw. (6.13)
Thus, if the winch is used to raise the sandbag a distance ∆xw then the gondola
is simultaneously pulled downwards a distance (mw/mg) ∆xw. It is clear that we
could use a suspended sandbag as a mechanism for adjusting a hot-air balloon’s
altitude: the balloon descends as the sandbag is raised, and ascends as it is low-
ered.
Our next example is pictured in Fig. 47. Suppose that a cannon of mass M
propels a cannonball of mass m horizontally with velocity vb. What is the recoil
m