1. 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):

    Δp = ∫ Fexternal dt
    A force is external if it involves any object not in the system. For colliding objects, the system consists of the objects undergoing collision.
    Assuming Fexternal 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:
    p1ix + p2ix = p1fx + p2fx
    p1iy + p2iy = p1fy + p2fy,
    m1 v1i cos θ1i + m2 v2i cos θ2i = m1 v1f cos θ1f + m2 v2f cos θ2f
    m1 v1i sin θ1i + m2 v2i sin θ2i = m1 v1f sin θ1f + m2 v2f sin θ2f.
    These are two equations in 10 variables; if energy is not also conserved, eight must be given to obtain a solution.
  2. 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.
  3. 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
    Fthrust = vexhaust dm/dt.
    Conserving momentum for a rocket (remembering that vexhaust is in the opposite direction of v), we have
    dp/dt = 0

    m dv/dt = vexhaust dm/dt

    v = vexhaust ln (m / m0)

  4. 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

    rcm ≡ ∫ r dm / M,
    where the integral is over infinitesimal pieces dm, M is the total mass and 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.


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

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