Lab Report #4 - Projectile Motion

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Experiment 4
Projectile Motion
by
Eugenio Panero
PHY 2091-06
experiment performed: Sep 13, 2006
report submitted: Sep 20, 2006
Lab Partner:
Chris Hubacek
Instructor:
Klaus Dehmelt
Introduction
The purpose of this experiment is to explore the mathematics behind projectile
motion. A ball with known, different initial velocities will be projected from a
known height. Measurements of the travelled distance will be taken, and
conclusions correlating experimental results with known formulas will be drawn.
Data
(See attached page)
Data Analysis
The distance travelled by an object with a non-constant velocity is given by the
formula:
βˆ’
1
π‘₯
=𝑉
2β„Ž
𝑔
0π‘₯
Where 𝑉
is the initial velocity, β„Ž
is the height, 𝑔
is acceleration due to gravity,
0π‘₯
and π‘₯
is the distance travelled. For the purpose of this experiment, the initial
velocity is not given by 0, meaning the object at rest, but as the velocity recorded
as soon as the object begins its projectile motion. A hypothetical ball that has an
initial velocity of 0.25 m sec-1 and falls from a height of 0.45 m will travel the
distance given by:
π‘₯
= 0.25 2 βˆ™
0.45 βˆ™
9.81 = 0.743 π‘š
Discussion
Diameter of the ball
Initial height of the ball
Trend line y-intercept
Trend line x-coefficient
Experimental value
2.9 cm ± 0.1 cm
100 cm ± 0.1 cm
-1.082
0.365
Standard error
0.398
0.003
Table 1: Summary of Results
Table 1 summarizes the experimental values obtained in the experiment. To
obtain a better understanding of experimental error, a theoretical data set should
be computed. This is done by applying the formula that has been presented in the
previous section.
Trial #
1
2
3
4
5
6
7
8
9
Velocity (cm / sec)
Theoretical Distance (cm)
181.250
161.111
152.632
138.095
120.833
87.879
72.500
54.717
38.667
Experimental Distance (cm)
81.8
72.7
68.9
62.4
54.6
39.7
32.7
24.7
17.5
Table 2: Theoretical and Experimental Distances
Percentage Error
64.7
58.8
54.1
49.7
42.4
30.9
25.4
18.7
12.7
20.94
19.17
21.50
20.29
22.29
22.13
22.41
24.31
27.26
A comparison between the theoretical and the experimental differences
emphasizes an important percent error: (20.03% average). Such an error is
beyond the limits of an experimental margin. The percent error will be visible also
when graphically representing the best fit line of the experimental and the
computational values.
Comparing Theoretical and Experimental Projectile Motion
Distance (cm)
90.0
80.0
70.0
60.0
50.0
Theoretical Distance
40.0
y = 0.4515x
30.0
20.0
Experimental Distance
10.0
y = 0.3649x - 1.0819
0.0
0.000
50.000
100.000
150.000
200.000
Velocity (cm / sec)
Table 3: Theoretical vs. Experimental Projectile Motion
The percent difference between the two slopes is given by:
π‘₯
βˆ’
π‘₯
0.4515 βˆ’
0.3649
𝑑
𝑒
100% =
100% = 19.18%
π‘₯
0.4515
𝑑
The percentage error between the two slopes is roughly equal to the average
percentage error between the ten measured distances. The precision of the
devices used for measurement cannot be the only cause for such an error. One
possible explanation is instead the effect that air friction has on the ball. This
would, however, imply that the theoretical results are incorrect. Verifying the
accuracy of the experimental results would, therefore, become impossible.
Considering the minimal amount of manual measurements performed, and the
sophistication of the apparatus used to measure initial velocity (millisecond
precision photogate), the intrinsic systematic error of ignoring air friction forces
may be the cause for the large discrepancy between experimental and theoretical
values.
Projectile motion is an example of an object having momentum. Momentum,
contrarily to what was believed with the Impetus theory, is not a residual force
left in an object. Momentum is the result of acceleration due to an impacting net
force, affected by the mass of the object itself. This is what is stated in Newton’
s
Second Law:
π‘š
π‘Ž
= 𝐹
Where π‘š
is mass, π‘Ž
is acceleration, and
𝐹
is net force.
Conclusion
Throughout the experiment, a mathematical correlation between the initial
velocity of the ball and the distance it travelled was found. Although it yielded a
relatively high percent error compared to its theoretical counterpart, it was
nonetheless sound in form.
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