EM375 Mechanical Engineering Experimentation
LAUNCHER SPEED and CONFIDENCE INTERVALS
BACKGROUND:
In this laboratory exercise you will fire ping pong balls from a launcher and measure their initial launch
speeds. From these data you will calculate the kinetic energy in the system immediately at launch. You
will use the results from this laboratory later in the semester for your final project.
PROCEDURE:
1.
Your group will be assigned a “stretch ratio.” This is the ratio of the length of the rubber tubing
immediately before release to the length of the tubing when it is unstretched (slack). It is imperative
that you record the unstretched length of the launcher tubes and the assigned stretch ratio, and include
these numbers in your report.
2.
Assign members of your group to the separate tasks: Arming the launcher and firing the balls;
retrieving the balls; and recording the data.
3.
Load a ball into the launcher and arm the timer. Release the ball, and record the speed indicated
on the timer. Select a variety of balls and continue firing them until you have recorded the speed for 40
launches. DO NOT eliminate any measured speeds. If the timer measures a speed, record it – even if you
“know” it is wrong. Only ignore a reading if you have a misfire.
4.
Measure and weigh a sample of the launcher’s rubber tubing, and weigh a launcher sabot
(pouch). Also weigh and measure 30 different ping pong balls. Record the measurements in the
provided data sheet.
5.
When you have finished, clean up!
DATA REDUCTION:
1.
Use the provided Excel file as a template. There are different tabs for: Speeds; ball data (weights
and diameters); and miscellaneous data. DO NOT sort the data. DO NOT eliminate any outliers.
2.
Still within Excel, convert all measurements from their units of measure into SI units. For
measured weights (lb or ounces) convert them to mass (kg). Your Excel worksheet should now hold all
the measured data, with the SI equivalent alongside.
3.
Still within Excel, calculate the mass of the rubber tubing per unit length (kg/m).
4.
Write a Matlab script that imports the first 10 speed measurements into Matlab and then:
a)
Plots the data set as a histogram. Use the MATLAB function histfit.
b)
Calculates the average launch speed, the median launch speed, the standard deviation,
and the 90%, 95% and 99% confidence intervals.
5.
Modify your Matlab script to import all 40 speed measurements, and then repeat step 2, above.
6.
Read all the ping pong ball mass data into Matlab and determine the average mass. Also read in
the rubber tubing mass per unit length, and then determine the total mass of all four of the launcher’s
bands combined.
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5.
Using the information provided in a separate handout, calculate the average total combined
energy in the ping pong ball and launcher at the instant the ball leaves the pouch.
REPORT REQUIREMENTS: NOTE: Your instructor may change this report requirement.
As a minimum, the main body of your report must include the following:
A statement of the stretch ratio you used.
A histogram of the entire (40 measurement) data set of launch speeds.
A single table that includes the statistics you calculated for the two data sets (10 measurements
and 40 measurements).
A second table that includes the different confidence intervals you calculated for the launch
speeds. Put the confidence intervals for the two data sets in adjacent columns in the table.
You should comment on whether or not the data appear to fit a normal distribution. Give
reasons to support your argument. Also include a discussion that explains the differences in the
confidence intervals you calculated, comparing the results for both data set size and level of
confidence.
A statement of the total kinetic energy in the system at launch for your specific stretch ratio.
Enclosures: At least the data sheet, a well annotated Matlab script file (use the publish function
in Matlab), and a copy of the lab handout.
PEER REVIEW:
After you have submitted your report to your instructor, you will be given someone else’s report to
review. You should go through their report in detail, checking and marking up format, sentence
structure, grammar, completeness, accuracy, ambiguity, etc. You will then complete the rubric, including
assigning scores.
Peer reviewed reports will be returned to the original author for rewrite and correction, before being
submitted to your instructor. YOU WILL BE REQUIRED TO SUBMIT YOUR ORIGINAL (peer reviewed)
COPY, THE COMPLETED RUBRIC, AND YOUR REWRITE.
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Total Kinetic Energy at Launch
There are three mass terms that have to be considered when considering the total kinetic energy at the
instant the ball leaves the sabot (pouch): The mass of the projectile (ping pong ball) itself; the mass of
the sabot; and a fraction of the mass of the tubing. This last term is because some of the tubing near the
pouch is accelerated to the full speed of the projectile, whereas some of the tubing (near the support
frame) is not accelerated at all.
Let’s see how to deal with the mass of the tubing. We assume that as the tubing moves, the speed of an
element of the tube is a linear function of its position down the tube. This means that, for example, if we
measure half way down the tube, the speed of that part of the rubber band will be half of the speed of
the sabot. Hence, the speed of the element is given by 𝑣𝑣 = 𝑉𝑉𝑂𝑂 𝑥𝑥/𝐿𝐿 where x is the position of the element
down the tube, VO is the instantaneous speed of the sabot at the instant the ball leaves the pouch, and L
is the unstretched length of the tube. The kinetic energy in an element of the rubber band will be:
𝛿𝛿𝛿𝛿𝛿𝛿 =
𝛿𝛿𝑚𝑚 𝑇𝑇 𝑣𝑣 2 𝛿𝛿𝑚𝑚 𝑇𝑇 𝑉𝑉𝑂𝑂2 𝑥𝑥 2
=
2
2𝐿𝐿2
where mT is the total mass of one tube and 𝛿𝛿𝑚𝑚 𝑇𝑇 =
kinetic energy equation and taking the limit yields:
𝑚𝑚𝑇𝑇
𝐿𝐿
𝛿𝛿𝛿𝛿. Substituting this mass equation into the
𝑚𝑚 𝑇𝑇 𝑉𝑉𝑂𝑂2 2
𝑑𝑑𝑑𝑑𝑑𝑑 =
𝑥𝑥 𝑑𝑑𝑑𝑑
2𝐿𝐿3
Integrate along the length of the tube to find the total kinetic energy in a single rubber tube:
𝐿𝐿
𝑚𝑚 𝑇𝑇 𝑉𝑉𝑂𝑂2 2
1 𝑚𝑚 𝑇𝑇 2
𝑥𝑥 𝑑𝑑𝑑𝑑 =
𝑉𝑉
3
2 3 𝑂𝑂
𝑥𝑥=0 2𝐿𝐿
𝐿𝐿
𝐾𝐾𝐾𝐾 = � 𝑑𝑑𝑑𝑑𝑑𝑑 = �
𝑥𝑥=0
We compare this result to the “normal” kinetic energy of a mass and see that the kinetic energy in one
tube of total mass mT can be calculated by assuming its effective mass is one-third of its total mass.
Remember that there are four tubes. Therefore the total kinetic energy in the system at launch is:
1
4𝑚𝑚 𝑇𝑇
𝐾𝐾𝐾𝐾 = �𝑚𝑚 +
+ 𝑚𝑚𝐶𝐶 � 𝑉𝑉02
2
3
where m is the mass of the ping pong ball, mC is the mass of the sabot and V0 is the launch speed of the
ping pong ball.
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