Kinetic Energy and Zero Momentum
What are at least two examples of an object or
collection of objects that have some kinetic energy, but a net
momentum of zero?
1) a game of "chicken", in which cars of the same mass and speed
drive directly toward each other.
2) two balls of identical mass connected by a spring and vibrating
toward and away from each other.
Most examples on earth are imperfect.
1) A ball and paddle connected by a string or rubber-band, tossed in the
air, watched by someone
who jumped exactly as high and fast as the assembly was tossed.
2) a solar system, watched by someone a few light-ears away, sitting
still relative to the sun.
3) a "Frisbee" spun rapidly in place and allowed to settle to the floor.
4) a baton, suspended from the ceiling by a string to it's center,
spinning end-around horizontal to the floor.
5) a plate spinning on the top end of a vertical stick
6) a rubber-band stretched across a light, stiff frame, then "twanged".
If there are many harmonics on band, then some part of the band is
7) a framework supporting equal and opposing pendulums,
of varying length so that at least one set is always moving.
8) a heavy stiff box containing many tiny bouncy rubber super-balls, well
9) a plastic Baggie full of room-temperature air. (We usually do not
consider heat as kinetic energy, but we could...)
10) any rotary machine tool (drill, lathe, router, lawn mover standing still)
11) the rear wheel on an upside-down bike, pedals cranked by hand.
I guess they all have some force-attachment loosely holding the group
They all lose their internal kinetic energy to the outside world at some rate.
And having zero net momentum is merely an accident
of being viewed from a velocity-matched frame of reference.
Two "ideal" billiard balls traveling at the same speed, colliding "head
on" is an example of a physical process that has zero net momentum but
non-zero kinetic energy. A more general way of approaching the question is
the following: The momentum of a collection of objects (particles), P, is
a VECTOR sum of the product of the masses, Mi, and the velocities, Vi. If
the SUM (Mi x Vi) remains unchanged before and after process -- that is,
the center of mass of the collection of objects does not change, then the
net momentum is zero. On the other hand, kinetic energy is a SCALAR sum of
the dot products of the masses and velocities of the objects: KE = 1/2 x
SUM (Mi x Vi * Vi). This can be non-zero even if the vector sum is zero.
The key to this problem is that momentum has direction, but kinetic energy
does not. If an object is moving, it has both momentum and kinetic energy.
For a set of objects to have kinetic energy, they must be moving. To have
zero momentum under these circumstances, the individual momenta must work
against each other, cancelling out one another.
The simplest example is two objects moving directly toward one another, each
with the same MAGNITUDE of momentum. Because they are moving, they have
kinetic energy. The total kinetic energy is the sum of the two. Because
the momenta are "equal but opposite", the sum of momenta is zero.
This can occur with any set of two or more objects. Find any set where the
momentum vectors add to zero and you have such a case. Another example is:
Two 500kg planes, each flying north at 200m/s.
One 2000kg plane, flying south at 100m/s.
If you add momenta together, you have:
100,000kg.m/s north, 100,000kg.m/s north, 200,000kg.m/s south.
The sum is zero.
Dr. Ken Mellendorf
Illinois Central College
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Update: June 2012