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Momentum
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times |
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or, |
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equivalently, |
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whose |
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time |
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rate |
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of |
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change |
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Motivation for Concept
Forces are actions which cause a change in the velocity of an object, but a given application of 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. Thus, although the change in velocity of a system is proportional to the force applied, it is not equal to the force applied. To define a quantity whose rate of change is equal to the force applied, we must include both the mass and velocity of the system subject to the force.
Mathematical Definition
Momentum of a Point Particle
The momentum (p) of a point particle with mass m and velocity v is defined as:
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to an object.{excerpt} ||Page Contents|| |{toc:indent=10px}| h2. Motivation for Concept [Forces|force] 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. h2. Fundamental Properties h4. Definition The momentum (_p_) of an object with mass _m_ and velocity _v_ is defined as: {latex}\begin{large}\[ \vec{p} \equiv m\vec{v}\]\end{large}{latex} h4. 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: {latex} |
Momentum of a System
For a system composed of N objects which are approximated as point particles with their position specified by the objects' centers of mass, the system momentum is defined as the vector sum of the momentum of the constituents:
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\begin{large}\[ \vec{p}^{system\rm \: sys} = \sum_{j=1}^{N} m_{j}\vec{v}_{j} \]\end{large} |
This definition is completely equivalent to
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\begin{large}\[ \vec{p}{latex} h4. Law of Interaction The rate of change of a system's momentum is equal to the vector sum of the forces applied to the object: {latex}^{\rm \: sys} = M^{\rm sys} \vec{v}^{\rm \: CM} \]\end{large} |
where Msys is the total mass of the system and vCM is the velocity of the system's center of mass.
Momentum and Newton's Laws
Momentum and Newton's Second Law
One way of stating Newton's Second Law is that the rate of change of a system's momentum is equal to the vector sum of the forces applied to the object:
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\begin{large}\[ \frac{d\vec{p}^{\rm \:system sys}}{dt} = \sum_{k=1}^{N_{F}} \vec{F}_{k} \] \end{large}{latex}
h4. Cancellation of Internal Forces
By [ |
Momentum and Newton's
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Third Law
By Newton's
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,
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cancel
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from
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above,
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leaving
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only
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contribution
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of
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}\begin{large}\[ \frac{d\vec{p}^{\rm \:systemsys}}{dt} = \sum_{k=1}^{N_{F}} \vec{F}^{\rm ext}_{k} \] \end{large}{latex} h4. Law of Change The change in momentum can be found explicitly by using the net external [ |
Momentum and Impulse
The integrated change in momentum can be found explicitly by using the net external impulse (Jext):
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impulse] (_J_^ext^): {latex}\begin{large}\[ \vec{p}^{\rm \:systemsys}_{f} - \vec{p}^{\rm \:systemsys}_{i} = \int_{t_{i}}^{t_{f}} \sum_{k=1}^{N_{F}} \vec{F}_{k}^{\rm ext} \:dt \equiv \sum_{k=1}^{N_{F}} \vec{J}_{k}^{\rm ext} \]\end{large} |
Conservation of Momentum
Conditions for True Conservation
In the absence of a net external force, the momentum of a system is constant:
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{latex} h2. Conservation of Momentum h4. Conditions for True Conservation In the absence of any net [external force], the momentum of a system is constant: {latex}\begin{large}\[ \vec{p}_{f}^{system\rm \:sys} = \vec{p}_{i}^{system\rm \:sys}\]\end{large}{latex} |
This
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up
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to
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and
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the
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components:
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}\begin{large}\[ \sum_{j=1}^{N} p^{j}_{x,f} = \sum_{j=1}^{N} p^{j}_{x,i} \]
\[ \sum_{j=1}^{N} p^{j}_{y,f} = \sum_{j=1}^{N} p^{j}_{y,i} \]
\[ \sum_{j=1}^{N} p^{j}_{z,f} = \sum_{j=1}^{N} p^{j}_{z,i} \]\end{large} |
Approximate Conservation in Collisions
Because the change in momentum is proportional to the impulse, which involves a time integral, for instantaneous events:
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{latex} h4. Approximate Conservation in Collisions Because the change in momentum is proportional to the [impulse], which involves a time integral, for instantaneous events: {latex}\begin{large}\[ \lim_{t_{f}\rightarrow t_{i}} \int_{t_{i}}^{t_{f}} F^{\vec{F}^{\rm ext} \:dt = 0 \]\end{large}{latex} |
For
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approximately
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instantaneous
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events
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collisions,
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it
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to
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approximate
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the
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external
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impulse
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as zero by considering a system composed of all the objects involved in the collision. The key to the utility of this assumption is that often during collisions 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 during the collision).
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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. It is also completely incorrect to say that the momentum of each object is conserved. Only the system momentum is (approximately) conserved. 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}Note that "dominated" and "negligible" are terms whose precise definitions depend on the accuracy desired in the results.{note} {warning}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.{warning} |