Experiment 6: Conservation of Energy and Momentum in a Collision
You will check for conservation of kinetic energy and
for conservation of momentum in a collision between
two ball bearings. One ball bearing is rolled down a
ramp. You let it fly off to find its speed at the end of
the ramp, indicated by how far away it lands. Where it
lands is recorded by it hitting a sheet of paper laid over
carbon paper. The ball bearing is then rolled down
again, this time with an identical ball sitting at the end
of the track. After colliding, they fall to the floor, again
landing on paper over carbon paper. The velocity of
the one ball as it reaches the end of the ramp is used to
find the energy and momentum just before the
collision; the velocities of the two balls just after they
hit are used to find final energy and momentum.
PROCEDURE: Measure the mass of both ball bearings. They should be identical. Tape together
four sheets of paper and lay them over four pieces of taped carbon paper. Lay this on the floor,
regular paper on top, with the plumb bob hanging over the middle of the shorter side. (“Plumb”
means vertical the same way that “level” means horizontal. The “bob” is the weight on the end of
the line.) Weight the paper down.
Mark the point under the plumb bob.
Adjust the screw at the end of the track which the target ball will sit on:
a. Get your eye exactly level with the countertop.
Then, adjust the screw so that its top is at the same
height as the bottom of the groove in the ramp. This
way, the incident ball will knock the target ball
cleanly off the screw.
b. Check that the screw is not leaning to one side; it
should look like a fatter extension of the plumb line.
c. The arm should be turned about 50° or 60° to the
side, otherwise one ball won’t be going fast enough
to miss hitting the arm as it falls. In a trial collision,
check that both balls land at least 10 cm from the
plumb bob (hopefully more). If necessary, readjust
the angle that the arm makes. Then, recheck the
screw’s height.
Once adjusted, be careful not to bump it; the arm can be kind of floppy. Remember where you
released the ball; it must be the same throughout the experiment.
Without the target ball in place, release the other ball at least half a dozen times. Don’t move the
paper the balls land on yet.
Put the target on top of the screw. (The screw has a dimple
for it to sit in.) Do at least another half dozen trials. Have
the instructor check the sheet with the dots before going
farther.
Once you have finished making data runs, you can turn the
paper over to see the dots better. Measure the distance from
the plumb bob to each cluster of impact points, as shown.
The amount of scatter in the dots shows you the uncertainty: For example, if the closest dot is 24.2
cm from the plumb bob, and the farthest is 27.6, record the distance as 25.9 + 1.7 cm. Also,
measure the angles θ1 and θ2. For example,
if "max θ", as shown below, was 60, and
"min θ", was 50, then you would record the
angle as 55 + 5. Also, measure the vertical
height of the fall, z.
Calculations:
Show how to set each
calculation up in the space provided. Also,
do not over-round: A zero used only to
show where the decimal point goes is not
significant. To have three significant digits, you need something like .00437 not .004.
Compute the time it takes for a ball bearing to fall to the floor: The vertical component of the ball’s
motion obeys
z = vzit + ½azt2
In this case, vzi = 0 because the ball is moving horizontally at the end of the track. Also az = g,
taking down as positive. With these substitutions,
z = 0 + ½gt2 Solving, we obtain
The uncertainty in t is very small; accurately enough we can call it zero.
Next, find vi, the speed the first ball has at the end of the ramp. Once it becomes a projectile, the
horizontal component of its velocity doesn’t change, so its speed at the end of the ramp equals its
average horizontal speed while flying: vi = di/t. From this, compute the kinetic energy of the first
ball before the collision. Find the uncertainties with the usual rules. Either leave uncertainties as
percents or convert as indicated by the blank; the choice was based on what you will need next.
Find v1 and v2 the same way you found vi. From them, find the system’s total kinetic energy after
the collision.
Now, check for conservation of momentum.
components:
Momentum is a vector, so think in terms of
- Find the magnitude of the momentum before the collision, pi. Assuming no uncertainty in
mass, pi will have the same percent uncertainty as vi. Then, write what its x and y components
are. ( ⃑ points directly along the x axis.) Include each component’s uncertainty, in units of
momentum, calculated from the percentage.
- Find the magnitude of each ball’s momentum after the collision, p1, and p2.
- Find the components of ⃑
and - signs.
and⃑ , showing how in the space provided.
Pay attention to +
- Finding the uncertainty in a vector’s components from the uncertainties in its magnitude and
direction was discussed in lab 2. By a similar procedure (approximating the uncertainty as a
differential without minus signs),
Uncertainty in p cosθ = |
|
|
|
Uncertainty in p sinθ = |
|
|
|
where dp = uncertainty in p and dθ = uncertainty in θ in degrees.
(“d” actually means a vanishingly small difference. It is being used not quite correctly here
for a difference which is small but finite.)
- Write the total final momentum, ⃑ , as a vector.
Conclusions: Within the uncertainty, was kinetic energy conserved? (Don’t be surprised if a little
energy was lost. Realistically, the collision is slightly inelastic.) Was the x component of the
momentum conserved? Was the y component of the momentum conserved?
Please put the ball bearings back in their box. They tend to get lost otherwise.
PHY 131
Experiment 6: Energy and Momentum in a Collision
DATA:
m = __________ + 0 (uncertainty small enough to ignore.)
di = ___________ + ___________
(percent uncertainty = ____________%)
d1 = ___________ + ___________
(percent uncertainty = ____________%)
d2 = ___________ + ___________
(percent uncertainty = ____________%)
θ1 = ___________ + ___________
θ2 = ___________ + ___________
z = ___________ + 0 (also small enough to ignore.)
COMPUTATIONS:
Compute t:
Just before collision:
vi =
= ___________ + ___________ %
KEi =
= ___________ + ___________J
Just after collision:
v1 =
= ___________ + ___________%
v2 =
= ___________ + ___________ %
KE1 =
= ___________ + ___________ J
KE2 =
= ___________ + ___________ J
KEf = ___________ + ___________ J
Calculate pi:
⃑ =(
+
) ̂+(
+
) ̂ kgm/s
p1 =
= ___________ + ___________ kgm/s
p2 =
= ___________ + ___________ kgm/s
p1x = ___________ + ___________ kgm/s
p2x = ___________ + ___________ kgm/s
p1y = ___________ + ___________ kgm/s
p2y = ___________ + ___________ kgm/s
⃑ =(
+
) ̂+(
+
) ̂ kgm/s