In your lab notebook, start on a fresh page and write the date of the lab, your name and your
partners’ names, the title of the lab, and the objectives section. As always, skip a line
between your written lines. Write the three relevant equations, and answer the prelab
questions (in the notebook, which I will assess during class).
You may cut and paste (typically tape works better than glue to hold papers in the notebook)
the equipment (new section: safety precautions) and procedure sections. Please notice that
there are blanks within the procedure section that need to be filled in during the experiment.
Set up the data tables for the data section. Complete all this prior to the start of lab on
Thursday.
The writeup to this lab is due Friday, January 29, though you may submit it early. The
writeup consists of a photocopy of your lab notebook pages for Lab 3, from the title page to
the conclusion section.
Physics 221
Lab 3: Predicting the range of a cannon
Objectives:
• Measurement of the muzzle speed of a fired projectile
• Calculations of the uncertainty of the muzzle speed
• Predicting a “reasonable” range of values for the range of the projectile, then comparing
against the measured range of the projectile
Introduction and relevant equations used:
Galileo Galilei was one of the first scientists to assert the independence of the horizontal and
vertical motions of a projectile; he noted that a dropped projectile takes as long to hit the
ground as a horizontally-fired one, though, obviously, the fired one travels a greater lateral
distance.
You will measure the muzzle speed of the projectile launcher using the photogate(s), then
predict a range of values for the range of the projectile. Simultaneously, you will measure
the actual range of the projectile.
In your lab write the following three equations that calculate:
• the muzzle speed of the projectile based on the photogate measurements.
• the predicted flight time of the projectile based on the height of the projectile launcher.
• the predicted range of the projectile based on the muzzle speed and the predicted flight
time of the projectile.
Prelab questions
1. What is the practical reason you are using photogate(s) in this lab, and not the sonic
rangefinder?
2. What major assumption are you making about the predicted flight time of the projectile?
How can you assess whether this is a good assumption? By this, I mean, if you really were
concerned about it, what other measurement could you make that would test this
assumption?
3. How are you setting up the photogate(s) to measure the time interval? In other words,
what event(s) trigger(s) the photogate(s) to start and stop measuring?
4. What assumption are you making about the trajectory of the projectile between the
photogates? How can you assess whether this is a good assumption? By this, I mean, if you
really were concerned about it, what other measurement could you make that would test this
assumption?
Equipment: (As always, note any deletions or additions)
Projectile launcher
Photogate(s)
Yellow plastic projectile
Steel ball projectile
LabQuest 2 datalogger
Carbon and copy paper
Goggles
Meter stick
Tilt-measuring app
Take a photo or make a sketch of your final setup with all measuring devices in place to
include in your lab notebook here.
Safety precautions:
Because yellow balls are being fired rapidly all over the room, everyone must wear the
goggles that are provided.
Procedure (work in teams of four):
1. Set the projectile launcher on the lab bench, and use the leveling app to make sure the
launcher is level (0° inclination). Record the vertical distance of the muzzle to the floor (in
m), so make sure the muzzle is hanging over the edge of the lab bench.
2. Set up the photogates. Use the ______ mode setting and the _____ time setting in the
base to measure the length of time it takes the projectile to pass through the photogate beam.
3. Begin with the yellow plastic projectile. Record the _______ and the _______ of the
projectile.
4. Make sure everyone in your group is wearing goggles, and that your group’s “line of fire”
is clear of other groups.
5. Carefully place the projectile in the launcher and use the loading rod to push the
projectile into the launcher. You should hear three “clicks” as the launcher spring
compresses.
6. Test-fire the projectile by pulling the trigger on the launcher. Do this without jerking the
launcher out of position. Make a note of where the projectile hits the floor; it may be a
considerable distance away. Mark the spot with tape and practice firing the launcher several
times to make sure the projectile lands more or less consistently near that spot and that the
photogate is getting a more or less consistent time interval measure of the projectile’s
passage.
7. When ready, place a piece of copy paper on the floor at the point where the projectile has
been hitting, and place a sheet of carbon paper, carbon-side down, on the copy paper. You
may use tape to prevent the papers from moving around.
8. Perform ten consistent launches. Record each time interval measured by the photogate on
the appropriate line in the data table. Label each of the carbon marks made on the copy
paper with the run number, so that you can tell one run from another.
9. After the runs are complete, remove the carbon paper and measure the horizontal distance
from the muzzle (you will have to project the muzzle tip onto the floor) to each of the spots
on the copy paper. Record each distance (in m) in the appropriate line in the data table.
10. Repeat the experiment with the steel ball projectile; again, the goal is ten consistent
launches.
11. Before the clean-up, do the preliminary calculation of the muzzle velocity (show the
calculation clearly in the next section) and informally compare this with your team-mates
and with other teams. The muzzle velocity should not vary greatly between teams.
Data section
Yellow plastic projectile:
Height of the muzzle: ________ m
_______ of the projectile: __________ (units?)
_______ of the projectile: __________ (units?)
Run number
1
2
3
….
Mean
Std. dev.
Time interval (s)
Muzzle velocity (m/s)
Projectile range (m)
Reality check: How many significant figures should each column have? What are the
instrument uncertainties for the time interval and the distance measurements?
A similar section should be developed for the steel ball projectile.
Analysis and results section
Assuming that the average muzzle velocity number is accurate, calculate the predicted
range your launcher should have had. Show the details of this calculation by copying,
calculating and filling in the table, using only the values for the height of the muzzle and the
average muzzle velocity (and the acceleration due to gravity):
Time (s)
Predicted xposition (m)
Predicted yposition (m)
0.00
0.05
0.10
…
Include as many rows incremented by 0.05 seconds as needed for the projectile to hit the
floor.
Calculate the percent error between your average measured range and the predicted range.
Show this calculation clearly.
Calculate the standard deviation for your average measured range.
Calculate the uncertainty of the predicted range of the projectile, which should involve a
particular instance of the uncertainty equation:
2
𝜕𝐶
𝜕𝐶
𝑢(𝐶) = √(𝑢(𝐴) ( )) + (𝑢(𝐵) ( ))
𝜕𝐴
𝜕𝐵
2
Graph (to be taped to this section of the lab notebook)
For the data in the table in the analysis and results section:
• Title this graph “Projectile trajectory – actual vs. theoretical.”
• You will need to decide where (0,0), the origin, of this graph is. There are a couple of
reasonable choices. Clearly plot the predicted x, y points in the table.
• Include a horizontal error bar for your predicted point of impact. This error bar size will
be the uncertainty of the theoretical range of the projectile.
• Include a vertical error bar for your predicted point of impact. This error bar is the
uncertainty in the height of the muzzle. Note that this error bar may be too small to show
effectively, in which case a note should be added to the graph about this.
• Use a smooth curve to connect the predicted data points.
• Using a different symbol or color, mark the average measured range and its error bar (the
standard deviation of the measured ranges) on this graph.
You may make the graphs by hand or use a graphing utility. If you do the latter, pay special
attention to making sure labels and units are on the axes, and that the data points show up
correctly plotted, then paste the printout into your notebook. In this case, it may be easier to
do the graph by hand, but do plot it on a piece of graph paper (ask the instructor if you do
not have any).
Discussion section
Question 5: (The usual “did you accomplish the goal of the lab” question) For each
projectile, state your average range and the uncertainty in its position, in the usual format
and matching precision between the numbers. Using the “standard deviations overlap” as
the criterion, is your actual range predicted by your theoretical range? Explicitly show the
comparison to justify this answer.
Question 6: Was there a difference in the measured range between the two projectiles?
Given that the launcher delivered the same force to each projectile, what factor accounts for
the difference in range?
Question 7: Calculate the percent difference between the average measured range of the
projectiles, using the formula:
𝑝𝑒𝑟𝑐𝑒𝑛𝑡 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 =
|𝑣𝑎𝑙𝑢𝑒 1 – 𝑣𝑎𝑙𝑢𝑒 2|
× 100%
𝑚𝑒𝑎𝑛 𝑜𝑓 𝑣𝑎𝑙𝑢𝑒 1 𝑎𝑛𝑑 𝑣𝑎𝑙𝑢𝑒 2
Then calculate the percent difference between the masses of the two projectiles. Comment
on whether these calculations are consistent or inconsistent with your answer to question 6.
Conclusion section
Question 8: Recall the two assumptions explicitly made in this experiment (questions 2 and
4). You were able to test one of these assumptions; comment about how that assumption
was tested, and whether, therefore, that assumption was reasonable.
Question 9: Would a smaller diameter projectile have led to a smaller, the same or larger
uncertainty in v? Do a semi-quantitative derivation of your answer.