Mass times velocity, or, alternately, a quantity whose time rate of change is proportional to the net force applied to an object.
Page Contents |
|---|
Motivation for Concept
Forces are actions which cause a change in the velocity of an object, but a given force will have very different results when applied to objects of very different mass. Consider the force imparted by a baseball player swinging a bat. When delivered to a baseball, the change in velocity is dramatic. A 95 mph fasball might be completely reversed and exit the bat moving 110 mph in the other direction. When delivered to a car, however, the change in velocity is miniscule. A car moving 95 mph will not be slowed noticeably by the action of a bat.
Fundamental Properties
Definition
The momentum (p) of an object with mass m and velocity v is defined as:
\begin
[ \vec
\equiv m\vec
]\end
Definition for System
For a system composed of N objects, the system momentum is defined as the vector sum of the momentum of the constituents:
\begin
[ \vec
^
= \sum_
^
m_
\vec
_
]\end
Law of Interaction
The rate of change of a system's momentum is equal to the vector sum of the forces applied to the object:
\begin
[ \frac{d\vec
^{\:system}}
= \sum_
^{N_{F}} \vec
_
] \end
Cancellation of Internal Forces
By Newton's 3rd Law, internal forces cancel from the vector sum above, leaving only the contribution of external forces:
\begin
[ \frac{d\vec
^{\:system}}
= \sum_
^{N_{F}} \vec
^
_
] \end
Law of Change
The change in momentum can be found explicitly by using the net external impulse (Jext):
\begin
[ \vec
^
_
- \vec
^
_
= \int_{t_{i}}^{t_{f}} \sum_
^{N_{F}} \vec
_
^
\:dt \equiv \sum_
^{N_{F}} \vec
_
^
]\end
Conservation of Momentum
Conditions for True Conservation
In the absence of any net external force, the momentum of a system is constant:
\begin
[ \vec
_
^
= \vec
_
^
]\end
This equation is normally broken up to explicitly show the system constituents and the vector components:
\begin
[ \sum_
^
p^
_
= \sum_
^
p^
_
]
[ \sum_
^
p^
_
= \sum_
^
p^
_
]
[ \sum_
^
p^
_
= \sum_
^
p^
_
]\end
Approximate Conservation in Collisions
Because the change in momentum is proportional to the impulse, which involves a time integral, for instantaneous events:
\begin
[ \lim_{t_
\rightarrow t_{i}} \int_{t_{i}}{t_{f}} F
\:dt = 0 ]\end
For approximately instantaneous events such as collisions, it is often reasonable to approximate the external impulse as zero. The key to such an assumption is if the change in momentum of any individual system constituent being analyzed is dominated by the internal collision forces (the external forces make a negligible contribution to that constituent's change in momentum).
Note that "dominated" and "negligible" are terms whose precise definitions depend on the accuracy desired in the results.
Before the collision occurs and after the collision is complete, the collision forces will usually drop to zero. Neglecting external impulse can only be justified during the collision.