Conservation of linear momentum
AP Physics C Mechanics requires that you be able to describe the behavior of a system using conservation of linear momentum. In its simplest form, conservation of linear momentum means the momentum of a system is constant: \[\begin{equation} \vec{p}_f = \vec{p}_0, \end{equation}\] or alternatively \[\begin{equation} \Delta \vec{p} = 0. \end{equation}\] However, there are some fine details we need to discuss.
Systems of particles, conservation of momentum, and center of mass
A collection of objects with individual momenta can be described as one system with one center-of-mass velocity.
For a collection of objects, the velocity of a system’s center of mass can be calculated using the equation: \[\begin{equation} \vec{v}_{cm} = \dfrac{\sum \vec{p_i}}{\sum m_i} = \dfrac{\sum m_i \vec{v}_i}{\sum m_i}. \end{equation}\]
The velocity of a system’s center of mass is constant in the absence of a net external force.
The total momentum of a system is the sum of the momenta of the system’s constituent parts: \[\begin{equation} \vec{p}_{sys} = \sum \vec{p}_i = \sum m_i\vec{v_i}. \end{equation}\]
In the absence of net external forces, any change to the momentum of an object within a system must be balanced by an equivalent and opposite change of momentum elsewhere within the system. Any change to the momentum of a system is due to a transfer of momentum between the system and its surroundings.
The impulse exerted by one object on a second object is equal and opposite to the impulse exerted by the second object on the first. This is a direct result of Newton’s third law.
A system may be selected so that the total momentum of that system is constant If the total momentum of a system changes, that change will be equivalent to the impulse exerted on the system \[\begin{equation} \vec{J} = \Delta \vec{p}. \end{equation}\]
Correct application of conservation of momentum can be used to determine the velocity of a system immediately before and immediately after collisions or explosions.
Selection of a system determines whether the momentum of that system changes
Sometimes they like to ask about how to define a system such that the momentum of the system is constant; or non-constant. This freaks out students, but do not panic. As someone doing a physics analysis you are allowed to pick system boundaries and control volumes as needed for your analysis.
Momentum is conserved in all interactions.
If the net external force on the selected system is zero, the total momentum of the system is constant, since \(\sum\vec{F} = \dfrac{d\vec{p}}{dt}\).
On the other hand, if the net external force on the selected system is nonzero, momentum is transferred between the system and the environment.
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If this sounds kind of abstract an academic, don’t worry. There are practical uses of it; for example, in fluid mechanics it is common to select control volumes where external forces are acting and the momentum of the system within the control volume is changing. Common also to define systems and control volumes where momentum is crossing into or out of the system. These usually enter into use later (e.g. during an introductory fluid mechanics class), but it is nice to know about them in an introductory physics class.
See also
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