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```									Projectile Motion

Projectile Motion

Pre-Lab

The pre-lab must be submitted to the lab instructor on a separate sheet of paper at the beginning of lab.

(1) A projectile, initially at height h0 above the ground, is fired horizontally and strikes the ground a
horizontal distance R from its launch point. Using the kinematics equations of motion, derive an
expression for the launch speed v0 of this projectile in terms of g, h0 and R.

(2) A projectile, initially at height h0 above the ground, is fired with an initial speed of v0 and at an angle of
θ above the horizontal. A certain target is lying on the ground. With what horizontal distance R must
the target be placed away from the projectile’s launch point in order for the projectile to hit the target?
Express your answer in terms of g , h0, v0, and θ.

Melvin J. Vaughn                              Page 1 of 6                          West Valley College
Projectile Motion

Concepts to Explore

•   kinematics
•   error analysis
•   statistical methods

Principles

Error analysis: Error analysis is the study and evaluation of uncertainty in measurement. Experience has
shown that no measurement, however carefully made, can be completely free of uncertainties. Since the
whole structure and application of science depends on measurements, it is extremely important to be able
to evaluate these uncertainties and to keep them to a minimum. In science, the word “error” does not
carry the usual connotations of “mistake” or “blunder.” “Error” in a scientific measurement means the
inevitable uncertainty that exists in all measurements. As such, errors are not mistakes; you cannot avoid
them by being very careful. The best you can hope to do is to ensure that the experimental errors are as
small as reasonably possible, and to have some reliable estimate of the experimental errors. We shall
use the term “error” exclusively in the sense of “uncertainty,” and treat the two words as being
interchangeable.

Commonly used formulas in statistical analysis
measured value - accepted value
% error                   % error =                              × 100%
accepted value
measured value 1 - measured value 2
% difference =                                         × 100%
% difference                           ⎛ measured value 1 + measured value 2 ⎞
⎜                                     ⎟
⎝                  2                  ⎠
Best estimate of quantity x based on a number of independent measurements:
N

∑x     i
Best estimate                                        xbest =   i =1

N
x ≡ xbest
1 N
Standard Deviation                                 σx =         ∑ ( xi − x )2
N − 1 i =1
σ
Standard Error                                       SEx = x
N
Fractional                                                                           σx
Fractional or relative uncertainty in a measurement x:
uncertainty                                                                            x
Error propagation
x ± y = x ± y ± σ xy where σ xy = σ x 2 + σ y 2
2          2
Error propagation                                                                ⎛σ    ⎞ ⎛ σy ⎞
x × y = x × y ± σ xy where σ xy           = ⎜ x   ⎟ +⎜
(multiplication)                                                               ⎜ x   ⎟ ⎜ y ⎟⎟
⎝     ⎠ ⎝    ⎠
may be treated statistically. i.e. Experimental uncertainties that can be
random uncertainty
revealed by repeating the measurements are called random errors.
systematic    cannot be treated statistically, i.e. it always pushes our result in the same
uncertainty    direction

Melvin J. Vaughn                             Page 2 of 6                            West Valley College
Projectile Motion

Projectile Motion: Projectile motion is the motion of a particle that is launched with an initial velocity v0 .
During the flight, the particle’s horizontal acceleration is zero and its vertical acceleration is the free-fall
acceleration due to gravity. If v0 is expressed as a magnitude v0 and an angle θ , the particle’s equations
of motion along the horizontal x-axis and vertical y-axis are:

vx = v0 x + axt , x = x0 + v0 xt ,            where vx = v cos θ
v y = v0 y + a y t , y = y0 + v0 y t + 1 a y t , where v y = v sin θ
2
2

Objectives

•   To distinguish between accuracy and precision.
•   To use statistical methods, including mean, standard deviation of the mean, and standard error,
to analyze the results of a set of measurements.
•   To practice identifying random and systematic errors.
•   To practice using kinematics equations to make predictions based on a physical model.

Apparatus

1.   spring gun
2.   “projectile”
3.   carbon paper
4.   plain paper
5.   measuring stick

Safety Awareness

Be aware of your surroundings when working with the spring gun and the projectile. The projectile may
move extremely fast and cause injury if it strikes someone. Make sure everyone in the vicinity of the
launcher is out of the projectile path and aware of each pending launch.

Procedure

NOTE: Today’s laboratory exercise will be lengthy. I recommend exercising good time management by
knowing when to perform your calculations and when to collect your data. Some calculations may be
done at home, if pressed for time. However, if you want to do the extra-credit activity at the end of the
lab, then you must have everything, including the calculations, completed.

Determine the launch speed:

The purpose of this first activity is for you to determine the launch speed of a fired projectile ball. This
value for launch speed will be used in the next activity.

1) Mount the spring gun horizontally at a vertical distance h0 above the surface that the ball will strike.

Question: Do you think it would be better to use a larger h0 or a smaller h0? Why? Discuss.

Melvin J. Vaughn                               Page 3 of 6                             West Valley College
Projectile Motion

2) In a trial run, fire the ball horizontally, observing the approximate location where the ball hits the
surface. Place a sheet of carbon paper on top of a sheet of plain paper and center the two sheets on
the point where the ball landed during its trial run. You will use these sheets to record and measure
the location where the ball hits the surface.

3) Reset the gun, being careful that it is positioned in a fixed location. Fire the ball horizontally a total of
ten nine times.

4) Measure and record the horizontal distance R from the ball’s launch point to where it hit the surface
for each trial.

5) Using your pre-lab derivation, calculate v0 for each trial. Use these values to calculate the ball’s
mean launch speed v0 . Calculate the standard deviation σ v 0 and the standard error SE v0 for the
ball’s mean launch speed. Report the best estimate of the ball’s launch speed for any trial as

v0 = v0 ± 2 ⋅ SEv 0

You will use this result for the next activity.

Hitting a target

The purpose of this activity is for you to determine where to place a target such that a projectile ball will
hit it.

1) Set up a target on the table top. We will call the table top the “ground.” It doesn’t matter if your target
is above or below the initial launch point of the ball, as long as it is not at the same height.

2) Elevate the launcher to an angle θ > 15° above the horizontal.

3) Using your pre-lab derivation, predict how far away (the range) you must place the target such that
the ball has a reasonable chance of hitting it. Your result should provide a best estimate for the range
by expressing it as

R predicted = Rbest ± σ R

where Rbest is your best estimate for the range and σ R is your estimate for the uncertainty in the range.

a) Question: Do you think the uncertainty in the ball’s launch speed will influence the uncertainty in
the ball’s range?

b) Question: What assumptions did you make in your predictions and how do you think they will
assumptions you’re making in your expectations, and how those assumptions may influence your
results.

4) Check your prediction to make sure that the values you used for h0 and θ are reasonable such that
you can successfully execute the experiment. (For example, does your set-up require that the
horizontal distance to the target be larger than the width of the lab room?) If the result is
unreasonable, then choose a different H or θ .

Melvin J. Vaughn                               Page 4 of 6                        West Valley College
Projectile Motion

6) Create a target with a bulls-eye at the center of several (at least three) concentric circles. The radius
of each circle, including the bulls-eye, should be an integer multiple of your estimate in the range’s
uncertainty, σ R , from above. For example, the bulls-eye should have a radius of 1σR, the first
concentric circle should have a radius of 2 σR,, and so on.

7) Place the target at the predicted spot. Shoot the ball. If the ball doesn’t strike near the target, then
recheck your measurements and calculations. (Hint: One of the largest sources of error is the
measurement of the launch angle and the initial height of the ball.)

8) Without moving the target, place a sheet of carbon paper and plain paper at a spot where the ball will
have a reasonable chance of striking it a number of times. This may mean that the carbon (and the
plain paper) do not line up with your bulls-eye target. Remember, the goal is to see how well your
predictions match with experiment, not whether or not you hit the bulls-eye. Use additional sheets of
paper to record where the ball actually lands in relation to the target.

9) Fire the ball a total number of 10 times.

10) Measure the horizontal distance from the ball’s launch point to each point where it hit the recording
paper.

11) Draw a sketch representing the target and the marks left by the ball’s impacts. Include this sketch as

12) Calculate the ball’s mean range R . Calculate the standard deviation       σ R and the standard error
SER in the range. Report the ball’s mean range as

R = R ± 2 ⋅ SER

Questions

1) Did the ball land at or around a consistent location, even if it wasn’t on your bulls-eye target? Does
the location of the ball’s marks indicate the presence of either random or systematic uncertainty?

2) Judging from your sketch of the impact marks left by the ball on the target, was your experiment
precise? Was it accurate? Explain.

3) What is the % difference between the ball’s predicted range and its measured range?

4) Is your experimental result consistent with your theoretical prediction? If not, why not? Discuss.
(See following Note.)

Note:

In lab, we will use the following criteria to determine whether an experimental measurement x is
consistent with the theoretical prediction xthy: If

x − 2 ⋅ SE x ≤ xthy ≤ x + 2 ⋅ SE x ,

then the experimental result and the theoretical prediction are said to be consistent. That is, if the
theoretical prediction falls within two standard errors of the mean, then the experimental result is
considered consistent with the prediction.

Melvin J. Vaughn                            Page 5 of 6                         West Valley College
Projectile Motion

5) What % of the balls landed within one standard deviation of the mean? Within two standard
deviations of the mean? Within three standard deviations of the mean? Is this what you would expect
from statistical theory (see “Confidence Intervals” in the Error Analysis packet downloaded for the first
lab)?

6) How do you think error and error propagation influenced your results?

EXTRA CREDIT CHALLENGE!

Note: You may attempt extra credit solely at the instructor’s discretion and only after you have completed
the experiment.

•   Ask the instructor to give you a target to hit.
•   You must provide a prediction for whatever quantity you need to determine in order to hit the
target.