The final shot in the pool game is yours, but the
cue and the eight ball aren’t nicely lined up with any of the pockets.
Looking around, the closest pocket is 45 degrees off the line between
the two balls. You take aim for a glancing blow, the cue ball strikes
the eight ball…what happens next? Do you sink the eight ball? Which way
does the cue ball end up going? And how can you make the eight ball go
off in a different direction?
Pool is a great example of physics in action. After every collision, the
momentum of
all the balls—the product of their mass and velocity—has to be
conserved. That is, the total momentum before the collision has to be
the same as the total momentum after the collision. And, roughly
speaking, the
energy must be conserved as well; the balls can’t fling away from each other with more energy than you give them. These two laws—the
conservation of energy and the
conservation of momentum—work together to steer the balls around the table.
In this project, you’ll experiment with colliding masses, see how
they collide, and maybe learn how to use physics to plan the perfect
pool shot!
At what angle will two equal-mass balls move away from one another after a glancing collision?
- 2 low friction masses of equal weight (hover pucks orair hockey pucks would work best)
- Smooth, flat surface (if using a pool table, try placing a foam board overthe surface to reduce friction)
- Protractor
- Tape
- String
- Place one puck on the surface and mark its starting position with the tape.
- Place the second puck a foot or so away from the first puck.
- Gently push the second puck towards the first puck, aimed so that
it hits the puck at a glancing angle rather than straight on (this may
take a few practice runs).
- Mark a couple of points along the paths both pucks tookafter the collision.
- Using the marks as a guide, lay a lengthof string along each of the paths taken by the pucks.
- Use the protractor to measure the angle between the strings—the angle at which the pucks moved away from each other.
- Repeat steps 1-6 several times and calculate the average angle
between the strings. If the puck doesn’t hit at a glancing angle, then
just skip that attempt and try again.
The angle between the pucks’ paths will be close to ninety degrees—a right angle.
In an
elastic collision, both momentum and kinetic energy are conserved. Momentum is given by
mvand kinetic energy by ½
mv2, where
m is mass and
v is velocity. If
vcrepresents the velocity of the moving puck before the collision,
vais the velocity of the moving puck
after the collision, and
vbis the velocity of the stationary puck after the collision, then conservation of kinetic energy leads to:
½mvc2=½mva2 + ½mvb2
Because all the masses are equal, the m’s cancel and you end up with:
vc2=va2 +vb2
This equation has the exact same form as the
Pythagorean Theorem,
c2 =
a2+
b2where
a and
b are the sides of a right triangle and
c is the hypotenuse. This only works if
vaand
vbare at right angles to one another.
The conservation of momentum adds some depth (and complexity).
Momentum is a little more complicated because it has to be broken down
into
components: the momentum along the original direction of motion (
x) and momentum perpendicular to that direction (
y). Momentum in both directions has to be conserved. Initially, all the momentum is in the
x direction:
mvcx = mvax + mv bx
Canceling the masses, you end up with
vcx = vax + v bx
In the
y direction, there is initially
no momentum. To make everything balance, that means the y-direction momentums after the collision must perfectly cancel:
mvcy = 0 = mvay + mv by
0 = vay + v by
vay = -v by
Putting this together with the conservation of energy, you find that
all the velocity components after the collision have the same magnitude,
with the
y components pointing in different directions. You end up with the final velocities pointing at right angles away from each other.
In an
inelastic collision, kinetic energy is
not conserved; some energy is lost to the surroundings. This means that, while the
ycomponents of the velocity still have to cancel, the
xcomponents can be different. The balls will no longer bounce away at right angles.
In reality, perfectly elastic collisions rarely happen; some energy
is always lost. Collisions between subatomic particles (protons and
electrons) are very nearly elastic; so are atoms in an ideal gas. Space
probes that slingshot around a planet behave the same way as elastic
collisions as well.
What happens if you use pucks (or balls) with different masses? For
example, if you’re using air hockey pucks, try making the stationary
puck be two pucks stacked on top of one another. What changes?
What happens if you use a surface that isn’t smooth (for example, a
carpet)? How does that change the angle? Is it less than or more than a
right angle? Can you explain what you’re seeing using the equations
mentioned above?