Lab #4
Prof. Susanne M. Lee
Projectile Motion
page 1
University at Albany, SUNY
Projectile Motion
Reading: Giambatista, Richardson, and Richardson – Chapter 4 (4.1 – 4.5).
Summary: Mark Twain wrote a book in which a Yankee from Connecticut in the 1800’s timetraveled back to King Arthur’s Court. The Yankee ended up being able to perform all sorts
of “black magic” because of his modern knowledge. In this lab you will be the Connecticut
Yankee and King Arthur needs your help to capture a neighboring walled town. To succeed,
you must determine at what angle to aim his cannon in order for the cannonball to hit in the
exact middle of the walled town. Once the cannon is in place outside the walled town, you
will have only one shot at getting the cannonball inside the town. However beforehand, in
the fields outside King Arthur’s court, you will be allowed to do a few experiments with the
cannon to characterize how fast the cannonball comes out of the cannon.
You need to design and perform the necessary experiments to determine the
“cannonball’s” initial velocity when fired from a spring-loaded “cannon”. Using this
information you will then determine at what angle the “cannon” will have to be aimed in
order for the “cannonball” to hit in the middle of the “walled town”. You will also learn
about data statistics and error analysis.
Note: do not spend more than two hours on this Pre-Lab. Go to one of the tutoring sessions
posted on the class web site (Course Organization folder) and get help.
Pre-Lab Analysis:
The answers to Question 1 contain all the general equations needed for the lab.
1.) In your pre-lab write-up, answer the following questions (from your textbook):
a.) If a “cannonball” is thrown upwards with an initial velocity of vi at an angle q to the
horizontal, what are the “cannonball’s” horizontal (vix) and vertical (viy) velocity
components in terms of vi and q? (10 pts)1
b.) In terms of vi, q, g, and t, what are the “cannonball’s” final horizontal (vfx) and vertical
(vfy) velocity components, if the “cannonball” was in the air for a time t? Don’t forget
that gravity acts only in the vertical direction. (12 pts)2
c.) What is the “cannonball’s” final vertical position (yf), if the “cannonball” starts at an
initial vertical position of yi? Leave your answer in terms of yi, vi, q, g, and t. (10 pts)3
d.) What is the “cannonball’s” final horizontal position (xf) if the “cannonball” starts at an
initial position of xi? (6 pts)4
In this lab, the spring-loaded “cannon” has three spring compression settings that fire
the “cannonball” at different speeds. You will choose one spring compression setting that
you will use for the whole lab and then determine what experiments to perform to measure
this initial speed. Since you will use the same “cannon” spring setting for the whole
experiment, every time you fire the “cannon”, the “cannonball” will come out with
approximately the same speed. It is approximate because the spring will not be compressed
precisely the same amount every time. Because of this experimental error, you will need to
do at least four trials to determine an average initial speed for the spring setting you chose.
The only equipment you will have for these experiments are a meter stick, the “cannon”, and
a “cannonball”.
Lab #4
Prof. Susanne M. Lee
Projectile Motion
page 2
University at Albany, SUNY
In the second part of this lab, you will need to determine at what angle to direct the
“cannon” so that the “cannonball” will land in the middle of the “walled town” you will be
given. The initial speed you found in the first part of the experiment together with these
Pre-Lab exercises will help you determine this angle. Then you will shoot the “cannon” and
see if you have successfully supported King Arthur’s expansion plans, i.e. whether your
“cannonball” falls – within experimental error – in the middle of the “town”.
The following questions will help you decide what sorts of experiments you might
want to perform when determining the initial “cannonball” speed.
2.) With the “cannon” aimed horizontally and set on the second compression setting, five
range measurements were made. The “cannonball” was found to hit the ground 1.15m
horizontally away from the end of the “cannon”.
a.) Sketch the “cannon” setup, making sure to identify the angle q and its value. Also
show the direction of the “cannonball’s” initial velocity vector and the “cannonball’s”
final position. [7 pts]5
b.) When the “cannonball” hit the ground, what was its final vertical position? (2!pts)6
c.) What was the “cannonball’s” initial vertical speed? (2 pts)7
d.) If the end of the “cannon” was 0.25m off the ground, how long did it take for the
“cannonball” to fall through this vertical distance and hit the ground? (8 pts)8
e.) When the “cannonball” hit the ground, what was its final vertical velocity? (Don't
forget to include the correct units.) (6 pts)9
f.) Find the initial horizontal speed of the “cannonball” as it left the “cannon”. (4!pts)10
3.) Now suppose there is a town whose precise center is 1.00m away from your “cannon”
and you want to shoot your “cannonball” into this center. If your “cannon” shoots a
“cannonball” with an initial speed of 5.0m/s, at what angle should you position the
“cannon” to accomplish this task? The questions below will help you through to the
answer. The “cannon” is now at the same height as the town center.
a.) Sketch this “cannon” arrangement, making sure to identify the angle q the “cannon”
makes with the horizontal. Also show the direction of the “cannonball’s” initial
velocity vector and the “cannonball’s” final position. [7!pts]11
b.) What are the horizontal (vix) and vertical (viy) components of this initial velocity in
terms of q and the initial speed? Don’t forget that the initial speed is 5.0m/s. [6!pts]12
c.) From the x-component of the velocity, how long will it take for the “cannonball” to
hit the ground in terms of q? Show all work. [12 pts]13
d.) In this amount of time, the “cannonball” rises to its maximum height and then falls to
its original vertical position, but displaced horizontally. Find q. Show all your work.
(Don’t forget that 2sinqcosq!=!sin2q) Hint: One way to proceed is to use the time you found
in part c in one of the projectile motion equations. (Note: you do not have to solve the
problem in this way, it is just something to get you started if you are really lost) [14
pts]14
4.) Summarize and outline the lab following the format posted on the Electronic Reserves
course web page. [20 pts]15
Lab #4
Prof. Susanne M. Lee
Projectile Motion
page 3
University at Albany, SUNY
Equipment to be used in this lab:
r Spring-loaded “cannon” with three different spring settings
r 1 metal “cannonball” (a small steel ball) – check this out from your TA. If you do
not return it at the end of lab, you will receive no credit for the lab.
r Meter stick and masking tape
r Plumbline (string with pointed weight attached at one end)
r Two blank sheets of paper
r 1 “walled town” (cardboard wrapped in a circle)
Experimental Procedure:
1.) Setup
r Make sure the “cannon” is securely clamped to the table (see Figure 1). If it is not,
notify your TA.
r CAUTION:
• Do not load the “cannon” until you are ready to fire it.
• Never put load the “cannon” until the path in front is clear.
• Do not place any body parts in front of any loaded “cannon”.
• Failure to maintain sufficient care to avoid injuring yourself and others is cause for
instant course failure and possible disciplinary action, whether or not any damage
actually results.
Figure 1: “Cannon” and “cannonball” (steel ball on the right). The yellow string is used to pull the
“cannon’s” trigger. The ball dangling from the end of the string on the left side of the “cannon” is the
plumbline used to read the “cannon’s” angle.
Lab #4
Prof. Susanne M. Lee
Projectile Motion
page 4
University at Albany, SUNY
2.) Determining the Important Physical Dimensions
r In your lab notebook, sketch the gun in relation to the ground and table, and label
it ‘ “Cannon” Sketch ‘.
• On your sketch, draw a coordinate system with your lab table at yf!=!0 and the
uncompressed position of the launching mechanism xi!=!0. [5 pts]16
a.) Locating xi in the laboratory
r Use the plumbline to determine the point on the tabletop corresponding to xi. Place
some tape on the tabletop over this point and mark on the tape the exact position
of xi.
• This is a little tricky:
1. Before doing anything, make sure that the font end of the “cannon” is
flush with its pivot point. If it is not, loosen the black plastic knob on the
other side of the mounting bracket and move the front end of the
“cannon” so it is flush with the black knob. Retighten the knob, making
sure to keep the “cannon” horizontal (plumb line should be lined up
with 0º).
2. Drop the other plumb line (not attached to the “cannon”) from the
“cannon’s” front end to the top surface of the “cannon” bracket and lay a
piece tape underneath the metal point of the plumb line. Mark on the
tape the precise position of the plumb line’s tip. Using a ruler, measure
how many millimeters this position is from the front edge of the
“cannon” bracket. Record this value in your notebook. (2 pts)17
b.) Measuring yi
r In a new sketch, show the path the “cannonball” will follow after it is launched.
On your sketch, label its initial position (xi!=!0, yi) and final position (xf, yf!=!0).
Title this drawing ‘Sketch of “Cannonball” Trajectory’. [5 pts]18
r Next, to find yi hold the plumb line from the top of the “cannon” barrel and place a
ruler next to the plumb line. This will ensure that you hold the ruler vertically.
r Record the distance from the tabletop to the bottom of the “cannon’s” barrel (yi).
Show this dimension on the your drawing. (3!pts)19
r Why is this dimension yi instead of the middle of the “cannon’s” barrel? Hint:
Think about which part of the ball hits the ground. Make sure you answer in
complete sentences. (3!pts)20
r Before proceeding, make sure that the front end of the “cannon” is flush with the
black pivoting knob. This way no height errors will be introduced when the
“cannonball” is shot upwards at an angle to the horizontal.
3.) Determining the “cannonball’s” initial speed
r Determine what set of experiments you should do in order to determine the
“cannon’s” initial speed. You may want to look back at the Pre-lab Questions for
help.
r Before doing any measurements, record in your notebook the steps you are going
to take. Show this to your TA and get his or her approval before starting your
measurements. (2 pts)21
Lab #4
Prof. Susanne M. Lee
Projectile Motion
page 5
University at Albany, SUNY
r Choose one “cannon” setting (1, 2, or 3) as marked on the side of the “cannon” and
record this number in your notebook. Make sure you label it clearly so that your
TA will understand the number. (2 pts)22
• Setting 1 range ≈ 130 cm
• Setting 2 range ≈ 170 cm
• Setting 3 range ≈ 230 cm
r Place the “cannonball” in the “cannon” and push the “cannonball” back until the
“cannon” locks into the position you have chosen.
r Fire the “cannonball” and perform the experiments you have chosen to determine
the initial “cannonball’s” speed. You should redo your experiments at least two
more times to see how much the spring compression varies each time you
compress it.
• When you shoot the “cannonball”, tape a piece of paper under the impact point, so you
can record each impact point more accurately. Mark each impact spot with a circle and
the corresponding Trial #.
4.) Analyzing your data
r Create a table in your lab notebook with columns for Trial #, xf-near, xf-far, and xf,
where the xf-near and xf-far are due to the width of the “cannonball” and xf is their
average. Don't forget to include units on all columns. Label it with an
appropriately descriptive title. (9!pts))23
r To measure xf-near and xf-far, you can use either a meter stick (make sure you measure
in a straight line) or if the ball landed farther than a meter away, you will need to
be more imaginative. One possibility is to use a taut string between the impact
point and xi. Describe in your notebook precisely how you measured each range
(use complete sentences). (6!pts)24
r Record your x values in the appropriate data table columns. (9 pts)25
r For each trial, average these two distances and record the value in the xf column of
your data table. This will give you the position of the impact point’s center. (6
pts)26
r Attach the paper with the 3 trial impact spots to your or your lab partner’s
notebook. Write both your names at the top of the paper and label the sheet
appropriately. [9 pts]27
a.) Calculating the initial (horizontal) speed
r Use Excel to determine the average initial speed of the “cannonball”. Hint: see
Prelab Question 2 for help in finding the initial speed of each xf you measured. The
Appendix to this lab will help you program Excel to calculate the average of all
your xf’s. (14 pts)28
r Print all of your Excel data and calculations. Attach them to a page in your lab
notebook. Make sure you show all formulas at the bottom of each calculation
column. (2 pts)29
r In previous years, the “cannon’s” spring mechanism was so reproducible that
every trial landed exactly on top of the previous one. If that is the case for your
experiments, then skip this part and go to the part that asks you to summarize
your results. If your data is not all exactly the same, then you can get extra credit
for calculating the speed’s standard deviation of the mean. Use a calculator and the
Lab #4
Prof. Susanne M. Lee
Projectile Motion
page 6
University at Albany, SUNY
formula for s given in the Appendix to this lab. Then find the 99% confidence
level of the standard deviation. Show all your work. Label and circle both
answers. (11!pts extra credit)30
r Summarize your results by writing the “cannon” compression # you used for your
experiments and underlining it. Next to this write “the mean initial speed =” and
then write the value you found (don’t forget units) ± the 99% confidence level of
the standard deviation – if you found the latter. Draw a big box around this
summary. (6!pts)31
5.) Predicting the “cannon” angle to hit in the middle of the “walled town”
r Your TA will place your “walled town” on your lab bench top a certain distance
away from the “cannon”. Find and mark the precise middle of your “walled
town”. At then end of this lab, you will turn in the “ground” (a piece of paper)
from the “walled town”. Describe in your notebook how you found the “town”
center. (8!pts)32
r The “walled town” is raised a height off the lab tabletop on purpose. This height is
precisely the same distance from the tabletop as the bottom edge of the
“cannonball” when it leaves the “cannon”.
r Accurately measure the horizontal distance from the end of your cannon to the
point you marked as the precise middle of your “walled town”. You will not be
able to measure this directly. Think how you will do this accurately and record
both the distance and how you measured it in your notebook. Make sure the
distance is clearly labeled. (10!pts)33
r Knowing this distance and the initial speed that your “cannon” will fire the
“cannonball”, determine the angle at which you will need to shoot the cannon in
order to hit the center of the “town”. Show all your work (Hint: look at Prelab
Question 3 and go through each part of that question but using the values you have just
measured.) [7!pts]34, [6!pts]35, [12 pts]36, [14 pts]37
• Make sure you label this value clearly and appropriately. Circle it. [3!pts]38
r When you did your previous trials to determine the “cannonball’s” horizontal
speed, if all your trials gave the same speed, then skip to Section 6. Otherwise,
again for extra credit, calculate a minimum and maximum range using the 99%
confidence level standard deviation you calculated in Part 5. Label and circle these
range values in your notebook. (8 pts extra credit)39
• On your sheet of paper, mark the minimum and maximum distances on either side of
your average impact spot.
6.) Do you get to remain in King Arthur’s Court? Firing the “cannon” at your predicted
angle
r Position the “cannon” at the angle you predicted in the previous section. Use the
protractor with attached plumb line (piece of string with a ball at the end) to set
this angle accurately. Assuming your previous measurements and calculations are
correct, everything now depends on setting this angle correctly, so do it very
carefully.
r While your TA is watching, fire the “cannon” three times. Circle each impact spot
and label it with the trial #.
Lab #4
Prof. Susanne M. Lee
Projectile Motion
page 7
University at Albany, SUNY
r Attach the paper with these 3 trials and your predictions marked on it. Write your
names on the paper and staple it to one of your lab notebooks. [11 pts]40
r Extra Credit: if your three positions are not identical, then do the following:
• Calculate the average position of the three trials. Circle and label your answer.
[6 pts]41
• Calculate the standard deviation of this average; circle and label your answer.
[6 pts]42
r How many millimeters away from the center of the “town” did the “cannonball”
hit? What percentage was this of the overall distance between the end of the
“cannon” and the “town” center. [6 pts]43
r The real King Arthur was renowned for his accuracy in shooting cannons. In order
to impress him, you have to have an accuracy of better than 1% or he will throw
you in the dungeon as being useless to him. From your experimental results,
should you be thinking of how to escape before being thrown in the dungeon or is
it okay for you to stick around for the promotion to head of King Arthur’s cannon
division? Don’t forget to explain your answer and answer in complete sentences.
[4 pts]44
r If you should be looking how to escape from the dungeons, what experimental
quantities could you or should you have measured more accurately to obtain
better results? Make some suggestions on how you could improve your
measurements. (5 pts)45
7.) Lab Clean Up!
r Clean up your lab bench so there are no papers or tape on the tabletop. [2!pts]46
r Close all computer programs. [1!pt]47
r Call your TA over to your lab bench and show him/her that you have cleaned up.
Then return the following items to your TA:
• Your “cannonball”,
• the support on which “walled town” rested,
• the “walled town”, without the paper in the middle – that should be included
with your lab report,
• your final lab report with all attachments. [4!pts]48
Projectile Motion
Lab #4
Prof. Susanne M. Lee
page 8
University at Albany, SUNY
Excel Statistical Instructions
1.) Mean or Average:
a.) Click on the cell in which you want to enter the “Average” formula and type an “=”
sign, then
b.) go to the "Insert" menu
c.) select Function
d.) select Statistical
e.) click on “Average” in the right hand column
f.) the word Average will appear in your cell with the appropriate formatting
g.) between the ( ) insert the cell letters and numbers that are to be averaged
h.) press Enter and Excel will calculate the mean for you.
2.) 99% Confidence Level of the Standard Deviation of the Mean
a.) The standard deviation (s) from the mean, requires knowledge of how much each data
point differs from the mean and how many total data points you have. Thus you first
need to find the mean of your data points, as shown above, before you can calculate
the standard deviation. The specific formula for finding the standard deviation is:
N
s=
Â
2
xi = the value of the individual data points
(xi - x)
i =1
N -1
,
where
x = the mean
N = the total # of data points
To enter this formula in Excel you will first need to find the mean (say you
calculate this in cell A16) of all your data points (say they’re in cells A1 through A15).
Then create another column where you find the square of the difference between each
data point and the mean. For the first data point, the formula would be “=(A1A16)^2”. Suppose these squared values are in cells C1 through C15 and you have 15
data points.
Then the above formula in Excel terminology would become
“=SQRT(SUM( C1:C15)/14)”, where 14 = 15 (# of data points) – 1.
b.) To find the 99% confidence-level of the standard deviation, multiply s by 3. Then 99%
of all experimental values will lie within the mean!±!3s.
c.) To use Excel to calculate the standard deviation of a set of points that are, for example,
in cells B1 through B8, type StDev(B1:B8).