You have probably discussed **momentum**
in an earlier physics course.
Momentum is a combination of the mass of a body
and its velocity,
just like kinetic energy.
There are two main differences between
KE and momentum, however.

- Momentum is a
**vector quantity**, whereas KE is a**scalar**. The momentum of a body has a direction as well as a size; the direction is just the direction of motion.That means that there are really three quantities involved in the calculation of momentum, not just one.

- The units are different: momentum involves the velocity of the
object raised to the first power, whereas KE involves
the square of the velocity. That means that the
standard SI units for momentum must not be Joules.
In fact, the units for momentum don't have a fancy
name ....
If, for some reason, we wanted to convert a momentum-like quantity into an energy-like quantity, we would have to multiply by a velocity-like quantity:

Let's do a few simple examples.

Q: I toss a rubber ball of massm = 0.2 kggently to the East atv = 5 m/s. What is the ball's momentum? Q: Crazy Ellie fires a 125-grain cartridge from her Smith and Wesson .357 Magnum handgun (one grain is approx 0.065 grams). The bullet leaves the muzzle of the gun atv = 440 m/stowards the East. What is the bullet's momentum?

Why bother with momentum?
One of the main reasons is that when objects
collide or interact,
then -- as long as there are no significant external forces --
**the initial momentum of the system is the same as the final momentum**.

Be careful: this applies to the sum over all the objects involved, not each object individually. In other words,

In fact, this equality must apply to each component of the
vector momentum, so we could also write an equation for
each component separately.
I'll use subscripts *i* to mean "initial" and
*f* to mean "final".

You may recall that when I discussed kinetic energy, I was careful to add lots of qualifiers to the statement "kinetic energy is SOMETIMES conserved in interactions between particles." It was only collisions between hard, compact balls -- the "elastic" collisions -- in which KE was conserved. For example, in this collision between two cars (click on the first picture for a very large movie), KE is definitely NOT conserved:

- both cars are motionless afterwards, so the final KE is zero
- a great deal of KE was dissipated in the process of deforming pieces of metal, breaking glass, and so forth

On the other hand, the total momentum of this pair of cars IS conserved:

- the initial momentum was zero in each direction -- why?
- the final momentum is zero in each direction

Let's work through a few simple examples of momentum in the non-relativistic regime, just to see how it can be used in practice.

A car of mass **m _{1} = 2000 kg**
moving at

Which of the following is the final velocity **v _{f}**?

- 20.0 m/s to the right
- 4.21 m/s to the right
- 2.67 m/s to the right
- 2.35 m/s to the right

A car of mass **m _{1} = 2000 kg**
rolling slowly at

Which of the following are the final velocities?

- car 5.0 m/s to the left, truck 1.53 m/s to the right
- car 2.0 m/s to the left, truck 0.93 m/s to the right
- car 1.0 m/s to the left, truck 0.53 m/s to the right

A car of mass **m _{1} = 2000 kg**
is driving at

Which of the following is the final velocity of the combined wreckage?

- (-19.3 m/s East, -12.5 m/s North)
- (-14.5 m/s East, -11.3 m/s North)
- (-11.3 m/s East, -15.3 m/s North)

If the angle **α** is drawn as shown,
how large is it?

- about 38 degrees
- about 128 degrees
- about 142 degrees

An **invariant quantity** is one that
has the same value, no matter which observer is making measurements.
The space-time interval

is an example of an invariant quantity.

Is kinetic energy an invariant quantity? Let's find out.

The Blue Man throws a ball (**m = 0.2 kg **)
at a speed of **v = 20 m/s** to the right.

Q: What is the momentum of the ball, according to the Blue Man?

But the Blue Man and the ball are all travelling across
the landscape at a speed of **w = 50 m/s** to the right,
as measured by the Red Men.

Q: What is the momentum of the ball, as measured by the Red Men?

- You might look at the notes to one of the other courses I teach, University Physics I for more details about kinetic energy and momentum in the non-relativistic regime.

Copyright © Michael Richmond. This work is licensed under a Creative Commons License.