A common mistake involving conservation of momentum crops up in the case of totally
inelastic collisions of two objects, the kind of collision in which the two colliding objects
stick together and move off as one. The mistake is to use conservation of mechanical
energy rather than conservation of momentum. One way to recognize that some
mechanical energy is converted to other forms is to imagine a spring to be in between the
two colliding objects such that the objects compress the spring. Then imagine that, just
when the spring is at maximum compression, the two objects become latched together.
The two objects move off together as one as in the case of a typical totally inelastic
collision. After the collision, there is energy stored in the compressed spring so it is clear
that the total kinetic energy of the latched pair is less than the total kinetic energy of the
pair prior to the collision. There is no spring in a typical inelastic collision. The
mechanical energy that would be stored in the spring, if there was one, results in
permanent deformation and a temperature increase of the objects involved in the
collision.
The momentum of an object is a measure of how hard it is to stop that object. The momentum of
an object depends on both its mass and its velocity. Consider two objects of the same mass, e.g.
two baseballs. One of them is coming at you at 10 mph, and the other at 100 mph. Which one
has the greater momentum? Answer: The faster baseball is, of course, harder to stop, so it has
the greater momentum. Now consider two objects of different mass with the same velocity, e.g. a
Ping-Pong ball and a cannon ball, both coming at you at 25 mph. Which one has the greater
momentum? The cannon ball is, of course, harder to stop, so it has the greater momentum.
The momentum p of an object is equal to the product1 of the object’s mass m and velocity v :
p = mv (4-1)
Momentum has direction. Its direction is the same as that of the velocity. In this chapter we will
limit ourselves to motion along a line (motion in one dimension). Then there are only two
directions, forward and backward. An object moving forward has a positive velocity/momentum
and one moving backward has a negative velocity/momentum. In solving physics problems, the
decision as to which way is forward is typically left to the problem solver. Once the problem
solver decides which direction is the positive direction, she must state what her choice is (this
statement, often made by means of notation in a sketch, is an important part of the solution), and
stick with it throughout the problem.
The concept of momentum is important in physics because the total momentum of any system
remains constant unless there is a net transfer of momentum to that system, and if there is an
ongoing momentum transfer, the rate of change of the momentum of the system is equal to the
rate at which momentum is being transferred into the system. As in the case of energy, this
means that one can make predictions regarding the outcome of physical processes by means of
1 This classical physics expression is valid for speeds small compared to the speed of light c = 3.00×108m/s. The
relativistic expression for momentum is
2
2
p = mv
1 −v / c . At speeds that are very small compared to the speed
of light, the classical physics expression p = mv is a fantastic approximation to the relativistic expression.
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