Physics 120
Winter 2014
Lab #3: Measurement Errors and Uncertainties with a Projectile Launcher
In this lab, you will be using a spring-powered projectile launcher to fire a small plastic ball into
the air. You will use two different experimental methods to measure the initial velocity of the
plastic ball: measuring the maximum height that the ball reaches, and measuring the total time
that the ball spends in the air. You will then test your measurement by using the velocity you
measured to calculate how far a ball launched at an angle will fly, and placing a target on the
floor at that distance. Extra points will be awarded for successfully hitting the target.
Types of Error: In the course of this lab, you will be asked to consider two different kinds of
errors that can creep into any measurement you make:
Random Error: Random errors are, as the name suggests, random. They are the result of
small differences in the experimental procedure from one measurement to another. If you are
measuring the length of an object, holding the ruler at slightly different points on that object is
a likely source of random error.
Systematic Error: Systematic errors are errors in measurement that occur due to some
problem with the measurement procedure that is exactly the same for every trial. If you are
measuring the length of an object, using a meter stick that is marked incorrectly is an example
of a source of systematic error.
Uncertainty: When reporting your final measured value in an experiement, you must also state
the likely magnitude of any measurement error, to reveal to the reader that the measurement is
consistent with a range of values. This is called the “uncertainty” and reflects the level of
accuracy. This is also, sometimes, called the “error,” but this does not mean that there is known
to be an error of that value, but rather that there is likely to be some error of that magnitude.
Things to Consider Before You Begin:
1) You will determine the velocity of the ball leaving the launcher in two different ways: by
measuring the maximum height reached by the ball, and by using a stopwatch to time the
ball’s flight. Which of these methods do you expect to give better results (i.e., smaller
uncertainty)?
2) When measuring the height, you will find that the ball reaches a slightly different height each
time. What are some possible sources for this random error?
3) You will measure the ball’s time of flight using a stopwatch. Consider your reaction time
(that is, how quickly you can press the stopwatch button), and how it affects the measurement.
Think of a way to estimate your reaction time. Do you think this is a random error, or a
systematic error? What methods can you think of to minimize and/or account for your
reaction time?
Apparatus: A spring-powered projectile launcher will be used to fire a small plastic ball into
the air. You will also need a stopwatch to measure the time of flight and a two-meter stick to
measure the maximum height reached by the ball.
The goal of the lab is to determine the velocity of the ball leaving the launcher. As a test of your
measurement, you will then use that velocity to predict the flight of the ball when the launcher is
used to fire the ball at a given angle.
IMPORTANT: There is a number marked on the projectile launcher that you have been given.
MAKE SURE YOU RECORD THIS NUMBER on your answer sheet. Each launcher is
slightly different from the others, with a slightly different initial velocity.
Firing the launcher: The launcher will be fired straight up into the air. Place the launcher on
the floor, and make sure the protractor reads 90º, and that the screws holding the launcher to the
mount are tightened down. Make sure that the launcher is not aimed at a light (launchers which
are mistakenly fired on the “high” setting will fire a ball into the ceiling hard enough to shatter
light bulbs), set the launcher to the “medium” setting (the second notch down), and fire the ball
by tugging sharply on the launch cord. When firing the launcher, be sure to hold it down with
your free hand. Do a couple of tests firing the launcher to get comfortable with it, and to see the
trajectory of the ball.
Measurement Procedure: Work in groups of three: one person works the launcher, one records
the time of flight, and one measures the maximum height. Rotate through the roles, so that each
of you gets to do each job for ten shots.
Instructions for each of the three roles:
Launcher: Load the ball into the launcher, set it to “medium,” make sure your partners are
ready, and then fire the ball into the air. You may want to count down (“3…2…1…go”) to
make life easier for the other two people in your group.
Time Measurement: Measure the time that the ball spends in the air, using the stopwatch.
Start the watch when the launcher is fired, and stop it when the ball hits the floor. Record the
time of flight for each of ten launches in your data table. Record your own measurements
only—your partners will record their measurements on their own sheets.
Height Measurement: Using a two-meter stick clamped to the lab table, record the
maximum height reached by the ball (in meters) for each of ten launches. You will need to
stand on the table in order to read the two-meter stick accurately. Remember that you will
need to account for the height of the bottom of the meter stick in order to determine the full
height above the floor.
After ten launches, switch roles so that each member of the group does each job for ten shots.
Analysis: To reduce the amount of random error, average the results of each set of trials. As
random errors are equally likely to cause measurements to be too large as too small, the
averaging causes some values that are too high to cancel with values that are too low.
4) Using Microsoft Excel, find the average of your ten values of the height, and your ten values
of the time of flight. Record your average values in the spaces indicated on the answer sheet.
Also record the average values obtained by your partners in the spaces indicated.
5) The presence of random errors means that the outcome of any one measurement will be
uncertain. We can measure the amount of variation in a given set of measurements with a
quantity called the “standard deviation” (Excel command: stdev({cell range}) ), which gets
the symbol (lowercase Greek letter sigma). Determine the standard deviation for your
height measurements (h), and your time of flight measurements (t), and record it in the
space indicated. Record your partners’ standard deviations in the space indicated
6) Uncertainties in experimental measurements are determined using a quantity known as the
“standard error,” which is equal to the standard deviation divided by the square root of the
number of measurements, N:

x  x
N
(N=10 in this case). The standard error quantifies the final amount of random error in the
averaged result, and decreases as the number of measurements increases.
For the remainder of this lab, and the rest of Physics 120, we will use the standard error as a
measurement of uncertainty. All measured values in Physics 120 labs should be presented
with some uncertainty, in the proper format. You should state the result of a series of
measurements as the average value, plus or minus the standard deviation, with the appropriate
units. The uncertainty should have only one significant figure, and the average value should
be presented with the same number of decimal places as the uncertainty. For example, if the
average of your height measurements is 2.035795 m, and the standard error is 0.005473 m,
you should report your result in the format: “2.036 ± 0.005 m.”
Record your average height and time, with uncertainty, along with your partners’ values.
7) Compare the three measurements of the height made by the members of your group. Do they
agree with each other to within the experimental uncertainty—that is, do the average values
measured by your partners fall within one standard error of your average value?
8) Compare the three measurements of the time of flight made by your group using the
stopwatch. Do they agree with each other to within the experimental uncertainty?
Determining the Velocity:
Using your average values for the time of flight and maximum height, you can now calculate the
initial velocity of the plastic ball as it leaves the launcher, using the equations for projectile
motion discussed in class.
9) Calculate the launch velocity using the average of your height measurements. Calculate the
uncertainty in the launch velocity based on the propagation of uncertainty in the height
measurement, as explained by your instructor.
10) In a new column, calculate the initial velocity of the plastic ball for each of the individual
heights that you measured, and then find the average of these velocities (Hint: if you’re using
Excel to record your data, you can do this very quickly). Compare this value to that which you
get using the average of your height measurements to calculate the initial velocity.
11) Find the standard error of the velocities from the individual height measurements. Compare
this value to the value you get from propagating the uncertainty of the average height.
12) Record your velocity value from the height measurements, and those of your partners, with
the appropriate uncertainty, in the space provided on the answer sheet.
13) Calculate the launch velocity using the average of your time measurements and propagate
the uncertainty in the average time measurement.
14) Calculate the launch velocity for each of the individual times and find the average and
standard error of these velocities. Compare this value to the value you get using the average
of time of flight.
15) Record your velocity value from the stopwatch measurements, and those of your partners,
with the appropriate uncertainty, in the space provided on the answer sheet.
16) Compare the measurements of the initial velocity obtained by the different methods. Do they
agree with each other to within the experimental uncertainty? Does one method that seem
better than the other, judging by the size of the uncertainty?
Hitting a Target:
Using the equations for projectile motion discussed in class, and whichever of your measured
initial velocity values you think best, calculate the distance that a ball fired from your launcher
would travel if fired at an angle of 20º above the horizontal. Set your launcher to this angle,
measure out the appropriate distance, and place a piece of paper on the floor at the position
where your calculation predicts that the ball will land. Fire the launcher, and see how you did.
Remember, you will get extra points for hitting the target, so calculate carefully!
Union College
Physics 120
Winter 2014
Name: _____________________________
Partners: ____________________ ___________________
Worksheet for Lab#3: Errors with Projectile Launcher
1) Which method do you expect to give the best results, and why?
2) What are some possible sources of random error in measuring the maximum height reached
by the ball?
3) Estimate your reaction time (in seconds)________________
Describe how you came up with that estimate:
Will your reaction time be a source of random error, or of systematic error?
Experimental Data:
Time
Trial
1
Time of Flight (s)
Height
Trial
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
Height (m)
Your Average time: ___________________
Your Average height: __________________
Partners’ times: ___________________
Partners’ heights: ________________________
____________________
Standard Deviation (t)_______________
________________________
Standard Deviation (h)______________
Partners’: ___________________
Partners’: ________________________
____________________
________________________
Standard Error (t): ________________
tavg ± t:_____________________
Standard Error (h): ________________
havg ± h:_____________________
Partners’: ___________________
Partners’: ________________________
____________________
________________________
7. Does your measurement of the average height agree with those of your partners within the
experimental uncertainty? Why or why not?
8. Does your measurement of the time of flight agree with those of your partners within the
experimental uncertainty? Why or why not?
9. Calculate the launch velocity using your average height value. As explained by your
instructor, calculate the uncertainty in your launch velocity based on the standard error of your
average height. Show your work in the space below.
10. Calculate the launch velocity for each individual height measurement.
11. Calculate the standard error of these velocity values.
Are your answers to 9-11 consistent?
12. Launch velocities with uncertainty based on average height measurement:
(your value): ____________________ Partners’ values: _______________________________
13. Calculate the launch velocity, with uncertainty (as explained by your instructor) using your
average time value. Show your work in the space below.
14. Calculate the launch velocity for each individual time measurement and the standard error of
these velocity values. How does the average and standard error of these values compare with the
calculated velocity and uncertainty in no. 13?
15. Calculate the launch velocity for each individual height measurement, average these results,
and calculate the standard error. How does your average launch velocity value and standard
error compare to your answer to no. 14
16. Launch velocity with uncertainty (your value): _________________________
Partners’ values: _______________________
_________________________
Do the two velocity measurements using the different methods agree with each other to within
the experimental uncertainties? Why or why not?
Which of velocity measurement do you think is better? Why?
On a separate sheet, calculate the range of the projectile d in terms of the initial speed, v, launch
angle , and initial height, y0.
Find the range for an angle of  = 20o: __________________
Did you hit the target?
Have your instructor write his/her initials here if so: