115L Lab OneUsing Physical Principles and Measurements to .docx

115L Lab One
Using Physical Principles and Measurements to Make a
Prediction:
Target Practice with the Ballistic Pendulum
1 Introduction
Physics is an important science largely because it allows us to
make accurate
predictions of objects’ behaviors in different situations. This
idea has been
applied in the engineering of buildings, vehicles, and energy
production. It is
used to design aircraft, plan space missions and execute battle
plans in warfare.
In the first part of this lab you will make use of two of the most
valuable
principles of physics, along with a couple basic measurements,
to determine the
speed of the ball launched by the spring gun in your ballistic
pendulum. In the
second part, you will calculate the point where the ball will
strike when it is
fired without the catching pendulum in place. You will fire the
gun to test the
accuracy of your predictions. Finally, you will do some simple
analysis of the
cause of any inaccuracy in your calculated targeting.

Tips for success:
• Make all your measurements as carefully as possible.
The more accurate your measurements are, the closer you will
come to hitting your target.
• Pay attention to units.
Calculations require that all units match for the numbers to
come
out right. For example, a distance may be recorded in meters,
centimeters, inches, miles, etc. The distance is fixed, but the
value of the number used to record it can vary greatly.
2 Using Conservation of Momentum and Con-
servation of Energy to measure the initial ve-
locity of the ball
Test fire your ballistic pendulum 3 or 4 times, observing the
parts and
how the mechanism works.
Be very careful not to get in the path of the ball!
Directly measuring the ball’s velocity as it is fired by the
ballistic pendulum
would be very difficult. However, since there are some physical
properties that
are conserved, meaning that the total amount cannot change–
only the form can
change or there can be a transfer from one object to another, the
ball’s speed
can be determined quite accurately with only a couple simple
measurements
1

and a couple short calculations.
How could you go about measuring the ball’s speed as it
launches from
the ballistic pendulum?
Why would it be hard to use that method with this equipment?
2.1 Tracking the Energy
The act of firing the ball from the ballistic pendulum, and the
different forms
that the energy involved takes during the process, can be viewed
in four distinct
steps:
• Loading
• Firing
• Collision between the ball and pendulum
• Swing of the pendulum
Loading. To load the ballistic pendulum, you have to push the
ball back
against a very stiff spring. In the process of doing this you use
some of the
energy stored in your body. Once the pendulum is loaded, the
energy that you
gave up is now stored in the compressed spring.It would be
difficult to make
a measurement of the energy leaving your body during the
compression of the
spring, but it is quite easy to put a number on the energy once it

is stored in
the spring. The energy depends on how stiff the spring is, and
how much it is
compressed. This can be written as a mathematical expression:
Energy stored in a compressed spring:
1
2
kx2 (1)
In this expression, k is called the spring constant. It quantifies a
property of
the spring–how “stiff” it is.The term x is the measure of how
much the spring
is compressed.
Measure the distance that the spring is compressed when you
load your
ballistic pendulum. Make sure you measure from a consistent
reference
point. Record the measurement here so that you can use it later
to
determine k for your spring.
How precise is your measurement? In other words, how much
could
your be off the actual distance–more or less than?
What creates this uncertainty?
So during the loading, the energy involved moves from storage
within the cells
of your body to storage in the compressed spring. Because of

this state of being
“stored,” both of these forms of energy are known as potential
energy.
As mentioned above, energy is a conserved quantity. The total
amount of
energy will always remain the same. It cannot be created or
destroyed. It can
be changed from one form to another. In practice, some energy
will always be
“wasted” during a transfer from one form to another. It isn’t
really lost, it just
ends up in some form, or forms, other than the intended one.
Some transfer
processes are more efficient than others.
The energy transfer during the loading of the ballistic pendulum
is from
a biologically stored potential energy to a mechanically stored
potential
energy. Where do you think some energy might escape to during
the
transfer?
(Think of the sweaty, grunting guy at the gym–maybe you are
the sweaty
grunting guy at the gym. Obviously a lot of energy goes into
lifting
several hundred pounds from one spot to a higher spot (another
form
of potential energy), but not all of it. Where does it end up?)
Firing. During firing, the energy is transferred from the
potential energy of the

compressed spring to the energy of the ball in motion. The
energy associated
with motion is called kinetic energy. The amount of energy
depends on the size
and speed of the moving object. Like the potential energy in the
spring, the
kinetic energy can be written mathematically:
Energy of an object in motion:
1
2
mv2 (2)
Here m is the mass of the moving object and v is its speed (v for
velocity).
Notice that the mass factor is in the expression in the first
power (called a linear
relation) and the speed factor is squared (quadratic).
If I’m lucky, I can throw a baseball about 45mph. A lot of major
league
pitchers can throw at 90mph. How many times more kinetic
energy
does a major league pitcher’s ball carry than mine?
The transfer of energy from compressed spring to moving ball is
actually quite
efficient, and now the principle of conservation of energy can
be put to use.
Fire the ballistic pendulum a couple times. Pay close attention
to the
first part of the process, just as the ball flies off the spring.
Since the potential energy and kinetic energy both have

mathematical expres-
sions describing them, and we’re looking at a process where
potential energy is
completely converted to kinetic energy, there is a way to write
the principle of
energy conservation itself as a mathematical expression. This is
done by setting
the two expressions (1 and 2) equal to each other:
Energy stored in the spring = Energy of the ball in motion
1
2
kx2 =
1
2
mv2 (3)
Since x and m can both be measured, this equation would allow
the calculation
of either k or v, if the other was known. But neither is,yet.
That’s why you’ll
need this next part...
Swing of the pendulum. The ballistic pendulum is set up so that
as soon
as the ball leaves the spring it embeds in the pendulum, which
swings upward as
it catches the ball. Partway up the arc there are some steps; and
the pendulum

has a ratchet lever so that it can swing out, but not back. This
means that the
pendulum will stop at the highest point in its swing. This
provides the opportu-
nity to make use of another form of potential energy, this one
easily measured.
Since gravity provides a constant force straight down to objects
everywhere on,
and slightly above, the surface of the Earth, any object
experiencing that force
has a potential energy stored. The higher up the object, the more
energy it
has available. You probably wouldn’t hesitate to drop your
backpack with your
laptop in it beside your chair as you sit down in class; but
chances are you
wouldn’t be willing to drop it off the third-floor landing just
outside the lab.
And, just like with kinetic energy, the amount of gravitational
potential energy
also depends on the size of the object involved. An acorn
dropping from an oak
tree onto a parked car will just bounce off and roll away. But if
the limb that
held the acorn drops–from the same height–hope they have good
insurance. The
mathematical expression is:
Gravitational potential energy: mgh (4)
The m stands for the mass again, and h is the height (above
some specific
point–you have to decide what point. It could be the floor in the
lab, it could
be the ground outside, it could be the table top. Gravitational
potential energy

is always relative to some other point). The g is the acceleration
provided by
the force of Earth’s gravity on any object. The acceleration is
the same for any
object. It actually depends on the distance from the center of the
Earth, but
near the surface of the Earth–the context of things in the lab,
and at the height
of people, buildings, trees, even airplanes and mountains–it is
okay to consider
it constant.
Now, by comparing the gravitational potential energy at the top
of the arc
to that at the bottom, you can say that the potential energy is
zero at the bot-
tom. Since nothing is moving once the pendulum sticks at the
top of the arc,
the kinetic energy there is zero. So the kinetic energy of the
moving pendulum
at the bottom of the arc is converted to gravitational potential
energy at the
top of the arc. This can be written as an equation:
Energy of motion at the bottom = Potential energy at the top
1
2
mv2 = mgh (5)
Fire the pendulum again. Pay close attention to the pendulum’s
motion
from stationary at the bottom to locked at the top of its arc.
The energy involved in this process follows the path from

potential energy in
your body, to potential energy in the compressed spring, to
kinetic energy of the
moving ball, to kinetic energy of the moving pendulum, and
finally gravitational
potential energy of the pendulum at the highest point of its arc.
The left side of equation 5 looks identical to the right side of
equation
3. If they were describing the same thing, you’d have the v that
you
need, by solving equation 5.
Why aren’t the two vs the same?
What else is different between the two kinetic energies?
What happens between the situation described in equation 3 and
that
described in equation 5 that hasn’t been considered yet?
The collision.Take a look at figures 1, 2, and 3. They show the
three stages
of the ball’s trip as the ballistic pendulum is fired. You’ve
looked at using
conservation of energy to write equation describing how
potential energy stored
in a spring is completely turned into the kinetic energy of a
moving ball, as
shown in figure 1. And you’ve looked at the equation describing
the process
shown in figure 3, where the kinetic energy of the swinging
pendulum becomes
gravitational potential energy as the pendulum ends up higher

than where it
started. However, the situation in figure 2 is a little different.
This is the exact
moment where the ball contacts the pendulum and becomes
embedded. Before
the collision the energy is in the form of the kinetic energy of
the moving ball;
after the collision the energy is still kinetic, now the moving
combination of
ball and pendulum. But the assumption that the kinetic energy
following the
collision is equal to the kinetic energy just before the collision
is not true here.
This type of collision, called “inelastic” because the two objects
stick together
rather than bouncing off each other, allows some of the initial
kinetic energy to
be converted into other types of energy not related to the motion
or position of
the objects.
One name for the form that energy takes during an elastic
collision is
“internal energy”.
Have you noticed anything different about the way the ball feels
after
a firing, or especially after a couple consecutive firings, that
would
indicate how internal energy might show itself?
Figure 1: Before the collision the ball carries kinetic energy that

was transferred
from the potential energy stored in the compressed spring. The
pendulum is
stationary, and at its lowest point.
Figure 2: At the moment of the collision the ball’s kinetic
energy becomes
kinetic energy of the combined ball and pendulum, but some of
the energy is
transferred to other forms. This is the step where conservation
of energy cannot
be used as a tool in the analysis of the process.
Figure 3: Just after the collision the ball and pendulum are in
motion with
whatever kinetic energy they are left with. As the pendulum
rises in its arc, it
slows down. Kinetic energy is transferred to gravitational
potential energy. At
some point, all the initial kinetic energy has become potential
energy and the
pendulum comes to rest. Conservation of energy is a good
approximation again
after the collision.
2.2 Collect the Post-Collision Energy Data
You will need to make use of a different principle of physics to
make the connec-
tion between the energy of the ballistic pendulum before and
after the collision,
and make the determination of ball’s speed leaving the spring.
But since con-
servation of energy is a good assumption during the pendulum’s

swing following
the collision, the gravitational potential energy at the end of the
swing will give
you a way to get the speed of the combined ball and pendulum
immediately
after the collision. The gravitational potential energy is easily
measured.
• Choose a reference point on that pendulum that will be easy to
use for measurements in both the lowest and highest positions.
• Measure the height of the pendulum at its lowest point. Enter
the
value in the table below.
• Fire the pendulum.
• Measure the height at the high point. Enter the value in the
table
below.
• The change in height for the pendulum, the difference between
the
high point and low point, is h. Enter this value in the table.
• Repeat these steps for five total trials.
Trial Bottom Top h=(Top-Bottom)
1
2
3
4
5
Since the height is not the same for each time the pendulum is
fired, the energy

given to the pendulum by the ball must not be the same each
time. You will use
the average change in height for the pendulum to calculate an
average kinetic
energy, and thus velocity, for the pendulum at the start of its
swing.
Calculate the average h here:
Using equation 5, 1
2
mv2 = mgh, and your average value for h, calculate
the velocity of the pendulum at the start of its swing. Show your
work
here:
2.3 Using Momentum to Find the Velocity Transferred in
the Collision
The idea that will allow you to determine the velocity of the
ball before the
collision is conservation of momentum. Momentum, like kinetic
energy, depends
on an object’s size and speed. But momentum also includes the
information
about the direction the object is travelling in. Technically,
direction information
is the difference between velocity and speed. The speed of an
object is how fast
it is moving. The velocity of an object is how fast it is moving
and in what
direction. The mathematical expression for momentum is just
mv, with m

being the object’s mass and v the objects speed in a certain
direction.
The really useful thing about momentum for calculations is that,
when you’re
considering motion in one certain direction, the total amount of
momentum in
that direction will always be the same. This applies even if
more than one
object is involved, as long as you remember to look at one
specific direction.
The principle will hold, even if there is an inelastic collision—
as with the ball
and pendulum in figure 2. The conservation principle means you
can write an
equation describing the collision: total momentum before = total
momentum
after. Using figures 1 and 2 as references for before and after,
respectively, the
conservation of momentum equation for the collision is:
Momentum before collision = Momentum after collision
mv1 = (m + M)v2 (6)
The left side has only the ball’s mass and velocity, since the
velocity of the pen-
dulum before the collision is zero. The right side has the
combined mass of the
ball and pendulum, now stuck together as one object, and the
new velocity of
that combination. Remember, the first motion of the pendulum
is in the exact
same direction that the ball was moving before the collision.
After it starts
moving the pendulum follows an arc, but that is because it is

attached to a
pivot and can’t continue in a straight line. The motion of the
pendulum along
the arc makes using momentum impractical for that part of the
process, which
is why you used energy conservation to analyse that portion.
But conservation
of momentum is just what you need to treat the exact moment of
the collision.
You will need the masses of the ball and pendulum. Use a triple
beam
balance to get the ball’s mass. The pendulum’s mass is marked
on the
apparatus.
Use equation 6, mv1 = (m + M)v2, to find the velocity of the
ball
before the collision, v1. Remember, v2, the velocity of the ball
after
the collision, is the velocity that you calculated using
conservation of
energy.
You also have the information now to find the spring constant,
k, using
conservation of energy.
Use equation 3, 1
2
kx2 = 1
2
mv2, to find k for your spring.

3 Using Physics and Calculation to Determine
the Ballistic Pendulum’s Target Range
Now that you’ve determined the ball’s initial velocity as it is
fired from the bal-
listic pendulum, you’re ready to take a look at a couple more
physical principles
that will allow a calculation to predict the point where the ball
will land when
fired without the catching pendulum in place. Then you’ll test
your prediction
by placing a target and seeing how close you come to hitting the
bullseye.
First you will need to make use of an idea very much related to
what was
discussed during the discussion of momentum. Remember,
momentum carries
information about a specific direction. When a ball is moving it
has momentum,
mv, in the direction is moving. In the direction perpendicular to
its motion,
its momentum is zero–because it has zero velocity in that
direction. Force is a
physical property that, like momentum, includes a direction.
Force is actually
very closely related to momentum. It is the change of
momentum.
Drop your ball from about the same height as the table.
What was its speed right before you dropped it?

What does that mean its momentum was?
How about right as it hit the floor?
Speed?
Momentum?
Now, this may seem obvious, but its important–what direction
was it
moving right as it hit the floor?
Remember, momentum includes direction. Was there a change
in the
ball’s vertical momentum after you dropped it?
Was there a change in the ball’s horizontal momentum?
The only force acting on the ball once you let go of it was due
to gravity. Gravity
is acting straight down, the ball’s momentum in the vertical
direction changed.
It sped up until it hit the floor. But nothing made its horizontal
momentum
change. It dropped straight down. The force only acts in one
direction—the
momentum only changed in one direction.
Drop the ball from table height again. Make an estimate of how
long it
takes it to hit the floor.
If the ball drops from table height to the floor this is how long
it will take it
to fall. This is true even if it has some motion in the horizontal

direction at
the start. The force due to gravity won’t cause any change in the
horizontal
momentum because it is a change of the vertical momentum–
perpendicular to
the initial horizontal motion.
Roll the ball off the edge of the table. Estimate the time it takes
it to
hit the floor.
Does it take the same amount of time as when you just dropped
it?
Repeat a few times with different speeds of rolling.
Can you tell any difference in the time it takes the ball to hit the
floor?
Does the speed of rolling change where the ball lands?
When you want to predict the location of an object at a certain
time, there is
an equation that is very useful. Again, the equation applies to
motion in one
specific direction. Since you’re interested in motion in two
directions—vertical
and horizontal—you’ll need to use the equation twice. The
equations are func-
tions of time. In other words, if you plug in a specific time, the
equations
will give you the location of the object at that time. It is
common to use x for
the horizontal direction, and y for the vertical direction. Here
are the equations:

Horizontal position:
x = x0 + v0xt +
1
2
axt
2 (7)
Vertical position:
y = y0 + v0yt +
1
2
ayt
2 (8)
Since you want to place your target where the ball will hit, and
you know the
vertical position where that will happen—the floor—take a
closer look at the
horizontal, x, equation first.
The general form for the equation for the horizontal position is
equation
7, x = x0 + v0xt +
1
2
axt

2.
• x on the left side of the equation is what you want to know. Its
your dependent variable.
• On the right side of the equation, x0 is the initial position of
the
ball. You’re interested in how far it goes from where it is now.
What’s a good choice for the value of x0?
• v0x is the ball’s initial velocity in the x direction.
In what direction is the ball launched by the ballistic
pendulum’s
spring?
• ax is the acceleration of the ball in the x direction.
Acceleration
is the change of velocity in a specific direction. It is very
closely
related to force being the change of momentum in a certain
direction.
What is your acceleration in the x direction?
• t is the time. It is your independent variable.
So, by calling your starting horizontal position zero, and
realizing that there is
no force (and thus no acceleration) in the horizontal direction,
the equation for
the horizontal position of the ball becomes:
x = v0xt (9)

With v0x being your measured value of the ball’s speed as it is
launched by the
spring.
How do you know what value to use for t?
You are interested in the moment the ball hits the floor, so the
length of time
from the release of the ball until the moment it hits the floor is
the t that you
need. Since the direction of motion from the table top to the
floor is vertical,
the information should come from the y equation. Take a closer
look at the y
equation, 8.
The general form for the equation for the vertical position is
equation
8, y = y0 + v0yt +
1
2
ayt
2.
• y on the left is the final position in the vertical direction.
What
will it be?
• On the right side, y0 is the initial vertical position of the ball.
You’ll need to measure this.
• v0y is the ball’s initial velocity in the vertical direction.

Does the spring launch give the ball any speed up or down?
• ay is the vertical acceleration. This is related to the force from
gravity. It is the same g that appeared in the expression for the
gravitational potential energy.
This equation takes direction into consideration–assuming you
measured up from the floor to the ball for y0, what sign should
you assign to g to account for the direction it acts in?
• t is the same t as in the x equation. If you know the rest of
parts
of the y equation, you can get t and solve the x equation.
If you haven’t done it already, measure the height of your ball
above
the floor, when it is loaded in the ballistic pendulum. This is
your y0.
Using your measurement for y0, solve the y equation for t:
0 = y0 − 12gt
2
Now that you know when the ball will hit the floor, you can use
the x equation,
equation 9, to find out where it will hit.
Use your t and v0x in the x equation to find where to place the
target.
x = v0xt
Carefully measure from the ball’s initial position out
horizontally the
distance x that you have calculated, and place your target.

Fire!
• Mark the spot where the ball actually hit, and measure the
distance it is away from your target’s bullseye.
• Shoot the ball a total of 5 times and get the distance that you
miss the bullseye for each.
• Get the average value for 5 “misses”.
One way to rate the accuracy of a measurement is to compare
the amount the
measurement is off from the actual value to the value of the
measurement. You
can do this by calculating what percent the distance of your
“miss” is of the
total distance you predicted.
Divide your average “miss” distance by the x you had
calculated, then
multiply by 100.
This is the percent of your prediction that you were off by.
An important part of using physics calculations to make
predictions is choosing
which factors must be considered, and which can be ignored.
The more details
that are considered, the more accurate the prediction will be;
but details also
increase the complication of the calculation. Often, something
that has a small
effect physically can be left out of consideration in the

calculation and the result
will still be accurate enough for the purposes. Physicists always
make choices
about what information to include in their calculations.
Were your “misses” long or short?
What factors that were ignored in setting up the calculations for
your
prediction might account for the trend in your misses?
Can you think of a quick adjustment you can make to the
apparatus
that will get you closer to the target?
Try it.
PS115
1st lab questions
1. If we assume that the ball is launched with the same velocity
every time from the spring, what can cause the maximum height
to be slightly different for each trial?
2. The pendulum experienced slightly different maximum
heights for each trial. In the experiment we took an average of
theses heights to get one initial velocity for the ball. Would
calculating a different initial velocity for each of these heights
and then taking an average of these velocities make more
physical sense then taking the average height?
3. If we were to change the spring constant (k) by increasing or
decreasing its value, what would the effect be on the initial

velocity of the projectile?
4. In what way if any would changing the angle of the gun
affect the projectiles expected flight distance and initial
velocity?
Challenge Calculation (you need to answer this question for
credit)
Assume that the pendulum was allowed to break off from its
lever arm when the ball hits it. Using the same masses, initial
velocity for the ball and initial height, how far would the ball
plus the pendulum travel if they hit at the edge of the table?

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