83
Experiment #2
Projectile Motion
January 20, 2006
Jane Doe
427 6666
Section X
Email: jd77@qlink
Lab partner: John Deere
APSC100 Module 2
TA: Isabelle Newton, Physics Dept.
ABSTRACT
An experiment was conducted to study projectile motion and calculate the
acceleration due to gravity. An inclined air-table and moving puck system was used in
which the puck was launched upwards and the motion recorded using a spark-timing
device. Results indicated that the system behaved as expected, with the horizontal
velocity component remaining approximately constant and the vertical velocity
component varying linearly with elapsed time. The value of gravitational acceleration
was calculated from experimental data to be g = 9.1  0.2 m/s2. This value did not agree
(within error) with the generally accepted value of g = 9.81 m/s2. This discrepancy was
likely due to additional drag forces that were not accounted for in calculations of the
experimental value.
I verify that this formal report is my own individual work and has not been copied
in whole or in part from another source (with the possible exception of equations,
tables and/or diagrams from the experimental descriptions on the APSC100-2
website). Furthermore, I have not and will not lend this report (electronic or
hardcopy) to any other student, either now or in the future.
Signed:__________________________
84
1.0
Introduction
A system displaying parabolic motion was produced by launching a puck up an
inclined, frictionless surface (air-table) as shown in Figure 1. The purposes of this
experiment were:
1. to investigate parabolic motion by measuring the x and y displacements as a function
of time, and use this information to determine x and y velocity variations, and
2. to experimentally determine a value for g, the gravitational acceleration.
z
L
Air-table
x
puck
h

y
Figure 1: Schematic diagram of puck being launched up a frictionless surface
air-table inclined at an angle The co-ordinate system is also shown.
For any object with constant acceleration, the general relationship between velocity,
v, time, t, and acceleration, a, is given by Equation 1:
v = vo + at
(1)
where vo is the initial velocity. This implies that the relationship between velocity and
time is linear, with the slope equal to the acceleration and the intercept equal to the initial
velocity. Figure 2 shows the xy plane of Figure 1, with the initial velocity of the puck an
angle  above the x-axis. The x and y displacements can be measured as a function of
time, and x and y velocities calculated from this information. Considering Equation 1,
acceleration in the x and y directions can be obtained from graphs of velocity versus
elapsed time.
y
vo

x
Figure 2: Projection of the air-table surface, showing the x and y directions, the
angle,  and the initial velocity, vo, of the puck when launched.
85
The gravitational acceleration, g, can be obtained from the acceleration of the
puck. Consider the schematic side view of the system as shown in Figure 3.
Z
y
N
m pg
Figure 3: Schematic (side view) diagram of



puck

showing
forces

If the surface is inclined at an angle , from Newton's second law the acceleration
of the puck in the x, y and z directions is respectively given by Equations 2, 3 and 4:
0 = mpax
- mpgsin = mpay
N – mpgcos = mpaz = 0
(2)
(3)
(4)
where ax, ay and az are the acceleration components of the puck, mp is the puck mass, g is
the acceleration due to gravity and N is the normal force of the surface on the puck.
Given experimentally measured values for ay, mp, and , a value of g can be
calculated from Equation 3.
2.0
Apparatus and Procedure
The apparatus and co-ordinate system used in this experiment are schematically
illustrated in Figure 1. A tube connected to the puck supplied a flow of air under the
puck to reduce the friction between the puck and table surface. The apparatus was
equipped with a spark timing device to record the position of the center of the puck on a
piece of paper resting between the puck and table surface. Sparks were produced every
0.1 s.
Initial measurements were made of the air table length L and height h (see Figure
1) in order to determine the angle . The experiment involved obtaining two different
traces for the puck. The first was a straight line trace, with the puck being released from
rest at the top of the table. This trace was used as a y axis. Without moving the paper, a
second, parabolic trace was produced by launching the puck upwards at an angle  from
one of the lower corners of the table. In order to estimate average puck velocity for each
spark point, displacement measurements in the x and y directions (x and y,
respectively) were made over 2 spark intervals - one on either side (i.e. t = 0.2s).
The displacements where measured using a ruler. The errors were assumed to be half of
the smallest ruler division (0.5mm).
86
3.0
Results and Analysis
The x and y components of the average velocity for each spark point were calculated
from the x and y displacements (vxx/t and vyy/t, respectively). Graphs of vx and vy
as a function of elapsed time are shown in Figure 4. It is clear from this data that both x
and y velocities vary linearly with respect to time.
0.400
0.300
Velocity (m/s)
0.200
v x = -0.0145t + 0.106
0.100
0.000
0.0
0.5
1.0
1.5
2.0
-0.100
-0.200
v y = -0.397t + 0.374
-0.300
-0.400
Elapsed Time (s)
vx
vy
Figure 4: Plots of x and y component velocities as a function of time for the parabolic
motion of the puck on the air-table. Lines of best fit are shown through the data points
along with the equations of each line.
Residuals graphs for the vx and vy versus time regression analysis are shown in Figures 5a
and 5b respectively.
87
Residuals
(a)
0.01
0.005
0
-0.005 0
-0.01
0.5
1
1.5
2
Elapsed Time (s)
Residuals
(b)
0.01
0.005
0
-0.005 0
0.5
1
1.5
2
Elapsed time (s)
Figure 5: Residuals graphs for a) vx vs time and b) vy vs time
The residuals show no obvious trend, indicating that the behaviour was linear as expected
and that the regression lines were a good fit to the data. As discussed in the introduction,
the slope of the velocity versus time curve gives the acceleration and the intercept is the
initial velocity. Table 1 shows the results of the regression analysis, which yielded the
values for acceleration and initial velocity (with errors) for the x and y component
directions.
Table 1: Values of acceleration (slope) and initial velocity (intercept) obtained from
regression analysis of x and y velocity plots in Figure 4.
Acceleration a
Acceleration
Initial velocity
Initial velocity
(slope)
standard error
vo (intercept)
standard error
2
2
(m/s )
(m/s)
a (m/s )
vo (m/s)
y component
-0.397
0.001
0.374
0.001
x component
-0.015
0.002
0.106
0.002
Finally, to find a value for g, the acceleration due to gravity, equation 3 was
rearranged to give:
 ay
g
.
(6)
sin 
From Figure 1 it can be seen that:
sin  = h /L.
(7)
88
Substituting this expression in equation 6 results in:
g
 ay L
h
.
(8)
The value of g determined from equation 8 and the data in Table 1 was 9.1  0.2 m/s2.
4.0
Discussion
Acceleration values determined experimentally are shown in Table 2.
Table 2: Summary of experimentally-determined accelerations
Acceleration (m/s2)
ax
ay
g
-0.015
-0.397
9.14
Standard Error
(m/s2)
0.002
0.001
0.22
Overall, this system did exhibit 'typical' projectile motion behaviour, since, as
shown in Figure 4, the x velocity was relatively constant while the y velocity exhibited a
linear dependence on elapsed time.
An interesting feature of this data was the non-zero value for ax, since ax is
usually assumed to be zero in projectile motion systems. The non-zero value for ax
implies that there was an additional force that was ignored in the original analysis in
section 1. This was likely to have been a drag force associated with the air hoses attached
to the puck.
Finally, a value of g = 9.1  0.2 m/s2 was determined from experimental
measurements. The accepted value for g is 9.81 m/s2 (Young and Freedman, 1997).
While these values are not in agreement within error, the non-zero ax value indicated that
there was a small, but not insignificant force that was not accounted for in the calculation
of the experimental value of g. Had this retarding force been taken into account, the
experimental value of g would likely have been much closer to the generally accepted
value of 9.81 m/s2.
5.0
Conclusions
In this experiment an inclined air-table/moving puck system was used to study
projectile motion. In general the system behaved as expected, with the x velocity
remaining almost constant and the y velocity varying linearly with elapsed time.
Furthermore, a value of the gravitational acceleration g was calculated from experimental
89
data. The calculated value of g = 9.1  0.2 m/s2 did not agree (within error) with the
generally accepted value of 9.81 m/s2. This disagreement was attributed to the fact that
the hose attachments on the puck produced an additional retarding force that was not
accounted for in the calculations.
6.0
References
[1] Young, H.D. and Freedman, R.A., University Physics with Modern Physics. Addison
Wesley, 1997.
7.0
Appendix – Data and sample calculations
Table A1 shows the data obtained from the spark paper, and calculated x velocities, vx,
and y velocities, vy, (with errors) for each point. Constant parameters are listed at the
bottom of table A1. The times (both elapsed and t) were assumed to have negligible
(zero) error. Measurement errors for x and y were 0.5mm, which was half of the
smallest ruler division.
Point #
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Table A1 – Projectile Motion Data
Elapsed x (m) y (m)
vx
vy
Time (s)
x/t (m/s) y/t (m/s)
0.100
0.200
0.300
0.400
0.500
0.600
0.700
0.800
0.900
1.000
1.100
1.200
1.300
1.400
1.500
1.600
1.700
1.800
0.0200
0.0205
0.0195
0.0198
0.0209
0.0195
0.0190
0.0200
0.0203
0.0190
0.0172
0.0180
0.0173
0.0170
0.0167
0.0164
0.0161
0.0155
0.0672
0.0585
0.0523
0.0430
0.0347
0.2780
0.0192
0.0109
0.0031
-0.0048
-0.0128
-0.0206
-0.0282
-0.0360
-0.0449
-0.0524
-0.0600
-0.0672
Δt (s) = 0.200
Error in Δx/Δt = 0.003 m/s (see sample calculation below)
Error in Δy/Δt = 0.003 m/s
Mass of puck = 569.6 +/- 0.1 g
Length of incline surface L = 585 +/- 2 mm
Height of incline h = 25.4 +/- 0.5 mm
0.100
0.103
0.098
0.099
0.105
0.098
0.095
0.100
0.102
0.095
0.086
0.090
0.087
0.085
0.084
0.082
0.081
0.078
0.336
0.293
0.262
0.215
0.174
0.139
0.096
0.055
0.016
-0.024
-0.064
-0.103
-0.141
-0.180
-0.225
-0.262
-0.300
-0.336
90
Calculation of errors for x and y velocities in Table 1
vx = x/t
  x   t  
The error in vx is
δvx = v x 

 but t) = 0
t 
 x
Inserting numbers from Point 1 in Table 1 and (x) = 0.5mm
0.0005m
 0.003m / s
δvx = 0.100m / s
0.020m
Since
The results of the linear regression in excel are as follows:
For vx vs t
Intercept
Slope
Standard
Coefficients Error
0.106471 0.001828
-0.01453 0.001689
For vy vs t
Intercept
Slope
Standard
Coefficients Error
0.374817 0.001284
-0.39741 0.001186
A value for g was determined from equation 6 using the acceleration of the puck
in the y direction determined from the linear regression (-0.397 +/-0.001 m/s2)
g
 a yL
h
=
 (0.397)(0.585)
0.0254
 9.14m / s 2
The error in g is:
 a y L h 
g  g 

 
a
L
h
 y
2
0.5 
 0.001


= 9.14 
 0.397 585 25.4 
= 0.2m/s2
The value of g obtained from experimental data is therefore 9.1  0.2 m/s2.
91
APSC 100: Formal Feedback
= yes, ok= fair, x = no, or needs improvement
Overall
Writing clear and language appropriate
(Colloquialisms, ambiguities, and clumsy phrasing avoided)
Information in each section is well organized and concise
Good understanding of experiment and results evident
(No incorrect or incoherent text)
Abstract
Purpose, method, results and significance succinctly summarized.
Final value(s) reported with uncertainty
Introduction
Objectives stated clearly
Theory developed logically, with relevant assumptions stated
All theory necessary to understand procedure and analysis presented
Equations numbered and variables defined when first used
Apparatus and Experimental Procedure
Clear, well-labelled, diagram(s) of the apparatus
Clear, concise description referring to diagram(s)
Concise summary of the procedure that would enable one to follow
what was done (paragraph form, past tense)
Results and Analysis
Analysis explained clearly
(Equations referred to by number)
Analysis complete
Graphs:
 Referred to and discussed in text
 Captions indicate content and context
 Axes labelled and have units
 Equations on regression lines with appropriate variables
 Error bars included
Results reported with uncertainties
(precision of results and uncertainty match)
Accepted values reported and referenced
Discussion
Did the results turn out as expected qualitatively?
How do the results compare quantitatively with theoretical or
accepted values? (Do they agree within error?)
Can unexpected results be explained by assumptions made in the
theory? What major sources of error predominated?
Answers to questions incorporated into discussion.
Conclusion
An overview of the experiment is given
Main results and conclusions are summarized
Appendix
Comments
92
Data and uncertainties in neat, titled, tables (units in header)
Sample calculations correct and complete.
Relevant Excel output included.
Mark at end of conclusion in report