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game. If we took the time to understand at least the basic physics of pool it might be
amazing to what degree we could improve our skills. Most of us already know at least
somewhat the general idea of how to play pool well. Below I will give a
brief description of how physics plays a part in improving you game of pool. So read on
if you care to impress your fellow pool players!
-Basic Momentum & Kinetic Energy
For the purpose of billiards we will not go into great detail as to what momentum
is. Basically though it can be thought of using the following equation;
p = mv
where p = momentum
m = mass of object
v = velocity of object
Kinetic energy is energy associated with the motion of an object. For basic purposes
we can just look at the following equation which relates kinetic energy with mass
and velocity of an object.
K = ½mv2
where K = kinetic energy
When you strike another ball with the cue ball it is almost a perfect elastic collision.
An elastic collision is one in which total kinetic energy as well as total momentum
are conserved within the system. This can be shown by the two basic equations;
Conservation of Kinetic Energy: ½m1v1i2 + ½m2v2i2 = ½m1v1f2 + ½m2v2f2
Conservation of Momentum: m1v1i + m2v2i = m1v1f + m2v2f
where m = mass of object
v = velocity
Since the cue ball has virtually the same mass as the other balls and the velocity of
our second ball will always be zero, since we are striking a static ball with the cue
ball. In addition this is considered a two- dimensional collision. From this we know
that momentum is saved within the y component and within the x component.
Therefore in the case of pool we can rewrite these two equations as:
Conservation of Kinetic Energy: ½m1v1i2 = ½m1v1f2 + ½m2v2f2
Conservation of Momentum: m1v1i = m1v1f cosø+ m2v2f cosØ
0 = m1v1f sinø - m2v2f sinØ
In this last equation the minus sign comes from the fact after the collision ball two
has a y component of velocity in the downward direction from the x-axis. This can
be seen in the following diagram.
The above diagrams show the initial velocity (both x and y directions) of both balls
(Vxi &Vyi) as well as the final velocities (Vxf & Vyf). As we can see Vxi = Vxf (total
of red and blue balls) as well as Vyi = Vyf.
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frictionless these equations are not exact due to a small loss of energy due to friction
but they are close enough that your game will improve if these ideas are taken into
In the game of pool each collision between balls is a close to perfect elastic collision.
From this we know that momentum as well as kinetic energy are conserved. By
knowing this it helps to figure out how much force to use when striking the cue ball
because we can do a quick calculation in our head to figure out approximately how
much of the velocity (depending on the angle) will carry over to ball two after the
collision. If you are hitting ball two at a sharp angle, you will have to strike the cue
ball harder in order for ball two to end up with the same velocity it would have if you
were to strike ball two with much less force at a small angle. Even though this sounds
a little confusing the diagram below and the diagram on the previous page might help
to explain. Practice of course will also make it much less complicated. In addition to
figuring out how hard to hit ball one, you also must know where on the ball to hit.
For straight shots the easiest way to determine where on the second ball you want the
cue to strike, is by mentally drawing a straight line from the center of ball two, to the
pocket where you are trying to sink it. Then where the straight line comes out on the
close side of ball two is where you want to aim for the cue ball to strike. This will give
the second ball the correct angle to make the pocket. In addition if you are trying to
set up your next shot, one can tell where the cue ball will end up by using the
equations for conservation of momentum.
As we can see that by changing the angle in which the cue ball strikes the other ball
we can directly change the direction and velocity at which ball two will leave the
collision. In addition one must keep in mind that a ball is not a particle, and therefore
it must be taken into account, the place where you are trying to strike the second ball
must line up with the side of the cue ball, not the center.
A combination shot is very similar to a straight shot except it involves at least three
or more balls. Now we have to consider the angle of the cue ball striking the second
ball so that it hits the third ball with the proper angle to fall into the pocket you are
aiming at. As in the straight shots momentum and kinetic energy are both
theoretically conserved, provided you are playing on a quality table. Again we must
remember when determining where to aim the cue ball that the side of the ball will
strike the ball first. This means your cue should be in line with the imaginary line
running from the point on the second ball you are trying to hit, and then running
through the center of the cue ball to the outside edge. This point on the outside edge
of the cue ball is where your cue should strike the cue ball. An example of this
thinking is shown in the diagram below.
Bank shots or kick shots, are when you hit the rail with the cue ball which then
bounces off the rail and hits the ball you are trying to sink. When trying to figure out
where on the rail to aim with the cue the thing to keep in mind is that the cue ball will
bounce back off the rail with the same angle it struck the rail with. Now since no pool
table is perfect the rail will probably cushion the cue ball changing it's angle slightly.
However the degree to which it will change the angle is in most cases very minimal
and therefore this is still an accurate way of lining up the cue. An example of this line
up is shown in the diagram below.
Some people find it easier to calculate the proper place to bank the cue ball off the
rail by imagining another ball in the line vertical with the ball you are trying to strike.
When the straight line from the cue ball crosses this vertical line is the location where
you place your "ghost ball." Where the line connecting the cue to the "ghost ball"
hits the rail is the point where you want to aim. This is shown in the following
With a greater velocity the less the rail will cushion the ball making the angles very
close to equal. In turn the slower the velocity of the cue ball the less accurate the cue
will bounce back with the same angle.