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- In the absence of external forces, the momentum of a system of objects is conserved, even when kinetic energy is not.
External forces can be incorporated into conservation of momentum much as they were in conservation of energy, by including a term giving the change in momentum delivered by those forces (called the

**impulse**):Δ

Assuming**p**= ∫**F**_{external}dtA force is external if it involves any object

*not*in the system. For colliding objects, the system consists of the objects undergoing collision.**F**_{external}is zero, the conservation of momentum equation looks deceptively simple. But consider a 2-dimensional collision involving just two objects (labeled "1" and "2"). Using subscripts "i" and "f" to denote initial and final quantities, and measuring angles relative to the positive x axis, the vector component equations are:p

and_{1ix}+ p_{2ix}= p_{1fx}+ p_{2fx}p

or_{1iy}+ p_{2iy}= p_{1fy}+ p_{2fy},m

and_{1}v_{1i}cos θ_{1i}+ m_{2}v_{2i}cos θ_{2i}= m_{1}v_{1f}cos θ_{1f}+ m_{2}v_{2f}cos θ_{2f}m

These are two equations in 10 variables; if energy is not also conserved, eight must be given to obtain a solution._{1}v_{1i}sin θ_{1i}+ m_{2}v_{2i}sin θ_{2i}= m_{1}v_{1f}sin θ_{1f}+ m_{2}v_{2f}sin θ_{2f}. - In an
**elastic collision**, kinetic energy is conserved. In an**inelastic collision**, it is not. In a**perfectly inelastic collision**, the colliding objects stick together. - For an object whose mass changes (such as a rocket), and assuming the exhaust velocity is constant,
we use Newton's laws to define the
**thrust**as**F**=_{thrust}**v**dm/dt._{exhaust}**v**is in the opposite direction of_{exhaust}**v**), we haved

**p**/dt = 0m d

**v**/dt =**v**dm/dt_{exhaust}**v**=**v**ln (m / m_{exhaust}_{0}) - The motion of an extended object or system of objects can be described in terms of the motion of its
**center of mass**, and the motion(s) of the constituents relative to the center of mass.The center of mass of an object or system of objects is

**r**≡ ∫_{cm}**r**dm / M,**r**is the vector position of each piece.If the center of mass of an object is not over a point of support, the object is unstable to tumbling.

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©2010, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.