Flying Through the Air
Wanted: Person with
background in
physics to design
exciting new circus
acts. Pay based on
performance.
Excellent benefits.
Apply in person to
the Personnel
Manager, Fair Lane
Circus, weekdays,
10 am - noon.
Introduction
While scanning the want ads in last Sunday’s paper you came upon this item. Intrigued, you went to
the circus and spoke to the personnel manager. She explained that Projil, the “Human Cannonball”
had an idea for a new act, but needed a person with experience in physics to convert the idea to
reality.
After a few moments, you were in Projil’s tent, where he explained his plan. The Ringmaster will
select a person from the audience who will choose the angle at which the cannon will be adjusted.
Then four assistants will move a landing net to the correct location, and Projil will be shot into the air
and come down into the net. The smaller the net, the greater the thrill. But, of course, if the net is
too small or in the wrong location, then Projil could end up in the hospital.
Your job would be to find out how large a net (target) would be needed and to tell the assistants
where to place the net. The smaller the net, the larger your pay. But, if Projil missed the net, not only
would you not be paid, but you had better be gone before he gets out of the hospital!
Knowing there was a projectile launcher in the physics laboratory that you could use to model the
circus act, you accepted the job.
Textbook Reading
Physics 150: Halliday, Resnick, and Walker: Chapter 4, Sections 5-6 (pages 59-64).
Physics 125: Cutnell & Johnson: Section 3-3 (pages 66-78).
Overview of Laboratory
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Flying Through the Air - Page 1 of 6
You and your team will first define the problem and analyze it theoretically. Then you will decide on
the properties of the system that you must measure. Next, you will design and conduct the
experiments needed to determine those properties. Finally, you will choose the size of the target you
are confident you can hit.
The laboratory instructor, like the person from the circus audience, will give you the launch angle. You
will have one chance to hit the target. The smaller the target, the larger the number of points you will
receive. But, if you miss the target, you will receive credit only for the theoretical analysis. Go for it!
You will receive up to 12 points for successfully analyzing the problem. If you hit the target within ±4
cm of your predicted distance, you will receive 4 extra points. Hitting the target within ±2 cm gets you
8 extra points, and if your projectile lands within ±1 cm of your prediction, you will receive not only the
full 20 points for the lab, but 4 bonus points as well!.
Objectives:
·
To solve an experimental problem involving projectile motion.
·
To design experiments to measure selected quantities.
·
To determine how variations in these selected quantities affect the desired experimental outcome.
·
To assess the reproducibility of an experimental result when many trials are made.
Pre-lab Questions
In preparation for measuring the initial speed of the projectile answer the following questions.
1. An object has an initial (unknown) vertical velocity vy0 . It rises with a with constant acceleration ay = –g.
After rising a distance h its velocity is zero. Write an equation giving vy0 in terms of g and h.
2. Do the Theoretical Analysis on page 3. First, do the pictorial and physical models. Then, do this step-bystep approach below to go the mathematical model. Put the correct equations in all blank boxes.
Strategy
Horizontal (x)
Vertical (y)
Identify the initial position
x0 = 0
y0 = h
Resolve initial velocity v0 at θ0 into x- and
y-components
vx0 =
vy0 =
Using your coordinate system, identify
components of acceleration.
ax =
ay =
Specialize the generic equation
x=
y=
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Flying Through the Air - Page 2 of 6
x = x0 + v0 t + ½at2 for x- and y-directions
Identify location where projectile lands
x=L
y=0
Find time when it lands in terms of L and
vx0 and substitute into equation for y.
t=
0=h+
Equation for y is now a quadratic equation
involving L. Solve it for L. Your
equation will involve g, h, L, v0, tan θ0,
and cos θ0.
Theoretical Analysis
Name______________________________
Partner _____________________________
A projectile is launched from at a given speed, v0, and angle, θ0, from a height h > 0 above the ground. It lands
on the ground a horizontal distance L from the launching point. Determine an expression for the distance L in
terms of the initial height, speed, and angle.
Pictorial Model
·
·
·
·
·
Draw coordinate axes.
Sketch the motion.
Identify x and y components of quantities
at key positions during the motion.
List known constants.
Identify unknowns that need to be
determined.
Physical Model
·
Draw the motion diagram for the projectile.
Mathematical Model
·
·
·
Write a general expression for the x and y components
of the position of the projectile as a function of time.
Specialize the expressions for the launch and landing times.
Combine the previous results to get an expression
for the distance L in terms of h, v0, and θ0.
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Check Result
·
·
·
What is L for a horizontal launch? How does
this distance depend on h and v0?
Does this seem reasonable? Why or why not?
What is L for a vertical launch? Does this seem
reasonable? Again, why do you say so?
When h = 0 your equation for L should become the
equation for the range of a projectile. Does it?
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Experiment Planning
Equipment
·
·
·
·
·
PASCO Projectile Launcher ME-6800 and ball
Meter stick
Lab stand and plate
Landing height box
Carbon paper
Introduction and Hints
Examine the projectile launcher. The launching angle can be adjusted by loosening the two thumb nuts
on the back of the upright. The angle can be read using the protractor and plumb line. The launch
position of the ball is shown. To launch the ball, insert it into the opening. Push it down with your
finger until the spring locks on the first catch (short range). Do not go beyond this first position. The
ball is released by pulling on the string attached to the release lever on the top of the launcher.
Measuring the Launch Speed, v0
There are several ways of measuring the initial (launching) speed of the ball, but in each case, you will
have to develop expressions to relate the initial speed to the distance the ball travels after its launch.
You may want to try more than one method so you can learn whether or not the launcher gives the
ball the same speed at all angles. For example:
·
·
·
You might launch it vertically. Place the lab-stand mounted plate directly above the launcher, and lower
the plate until you can just hear the ball hit the plate. Record methods and data on page 5.
You might launch it horizontally and place a sheet of paper under the carbon paper to record its landing
location on the lab table. Record methods and data on page 5.
You might launch it at a fixed angle and have it land on the wooden box that was constructed so that the
initial and final heights of the trajectory are the same. Record methods and data on page 5.
Testing the Theoretical Predictions
Once you have found the initial speed, you will want to test the predictions of your theoretical analysis.
Does it give the correct result at several angles? How reproducible are your results? That is, for a
given value of θ0, what is the uncertainty in L. Decide, on the basis of your tests, whether to go for 4,
8, or 12 points.
When you are ready, ask the laboratory instructor to give you your launch angle and indicate the target
size you will be attempting. You will be given two minutes to predict the place the ball will land.
Report
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The instructor will record how many points you achieved for your launch (0, 4, 8, or 12). Hand in your
theoretical analysis and a well-organized and well-presented record of the experiments you did to
measure the initial speed and the tests of your predictions. See page 5 for a model of a suitable
experimental record.
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Finding the initial or launch speed (v0)
Methods used (e.g. vertical, horizontal, fixed angle):
Measurements (e.g. heights reached, launch height and horizontal distance traveled, angles and ranges):
Calculated initial speed:
Assumptions and approximations made
For example: no air resistance, launch speed doesn’t depend on angle (did you test this?)
Comparison between experiment and prediction (theoretical analysis)
For example, you might develop a table of launch angle (θ0), predicted value of L, and measured values(s). You will
probably want to calculate, for each angle, the average value and the standard deviation or other measure of the spread in
distances. You could also produce a graph of the results.
Evaluation of Experiment
Summarize the results, comparing predictions and measurements in a way that allows you to know where to place the
target when the instructor assigns you the launch angle.
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