University Physics I - Classical Mechanics, 2019

3.4. IN SUMMARY 61
3.3.2 Recoil and rocket propulsion
As we have just seen, you cannot alter the motion of your center of mass without relying on some
external force—which is to say, some kind of external support. This is actually something you may
have experienced when you are resting on a very slippery surface and you just cannot “get a grip”
on it. There is, however, one way to circumvent this problem which, in fact, relies on conservation
of momentum itself: if you are carrying something with you, and can throw it away from you at
high speed, you will recoil as a result of that. If you can keep throwing things, you (with your store
of as yet unthrown things) will speed up a little more every time. This is, in essence, the principle
behind rocket propulsion.
Mathematically, consider two objects, of masses m 1 and m 2 , initially at rest, so their total momentum
is zero. If mass 1 is thrown away from mass 2 with a speed v 1f , then, by conservation of
momentum (always assuming the system is isolated) we must have
0=m 1 v 1f + m 2 v 2f (3.12)
and therefore v 2f = −m 1 v 1f /m 2 . This is how a rocket moves forward, by constantly expelling mass
(the hot exhaust gas) backwards at a high velocity. Note that, even though both objects move, the
centerofmassofthewholesystemdoesnot (in the absence of any external force), as discussed
above.
3.4 In summary
1. The inertia of an object is a measure of its tendency to resist changes in its motion. It is
quantified by the inertial mass (measured in kilograms).
2.Asystemofobjectsiscalledisolated (for practical purposes) when there are no net (or
unbalanced) external forces acting on any of the objects (the objects may still interact with
each other).
3. When two objects forming an isolated system collide in one dimension, the changes in their
velocities are inversely proportional to their inertial masses:
Δv 1
Δv 2
= − m 2
m 1
This may be used, in principle, as a way to define the inertial mass operationally.
4. The inertial mass thus defined turns out to be exactly (as far as we know) proportional to the
object’s gravitational mass, which determines the gravitational force of attraction between it
and any other object. For this reason, most often we measure an object’s inertial mass simply
by weighing it.