PHYS 163
Lab #7 Projectile Motion
ANALYSIS
From Procedure Parts 1 - 3
Dx (m)
±
Dy (m)
±
2.18
0.035
-1.1
0.015
q (deg)
sin( 2q )
R (m)
±
15
0.50
1.56
0.03
1.54
1.59
25
0.77
2.12
0.04
2.07
2.16
35
0.94
2.51
0.04
2.54
2.47
45
1.00
2.65
0.02
2.67
2.62
Using Equation 7.3 combined
with Parts 1 - 3 results
v0 (m/s)
±
v0 (H)
4.726
4.62
0.106
v0 (L)
4.514
Using Eq. 7.6 combined with
graphical results
v0 (m/s)
±
Slope
2.163
4.60
0.137
Slope(H)
2.291
Slope(L)
v0 (H)
2.034
v0 (L)
4.465
From Procedure Parts 4, 5
Steep
Shallow
RESULTS: v0
% difference of v0 values:
Do values agree?
0.345
yes
4.738
PHYS 163
Lab #7 Projectile Motion
DATA
Distances should be recorded in meters
Part I. Spring Gun on Table (Procedure Parts 1, 2, 3)
ybt = vertical distance from bottom of ball to tabletop
ytf = vertical distance from tabletop to floor
yfs = vertical distance from floor to shelftop of box
x0 = horizontal distance from tip of gun in unloaded position to front of table
xtb = horizontal distance from front of table to front of box
xi (i = 1, 2, 3) = distance from front of box to "ith" mark
Dy, Dx represent displacements
Distance
±
ybt
32.1
0.5
ytf
90
yfs
13.5
Dy
-108.6
|xi-xavg|
Distance
±
x0
0
0.5
0.5
xtb
200
1
0.5
x1
16.5
0.2
1.0
1.5
x2
19.5
0.2
2.0
x3
16.6
0.2
0.9
xi,avg
17.5
0.2
Dx
217.5
Part II. Spring Gun on Floor (Procedure Parts 4, 5)
x0 = distance from tip of gun in unloaded position to front of box
xi (i = 1, 2, 3) = distance from front of box to "ith" mark
R represents horizontal displacement when Dy = 0
15o
x0
Distance
±
198
1
x1
20
0.2
25o
x0
Distance
±
233
1.5
1.3
x1
14
0.2
3.6
|xi-xi, avg|
|xi-xi, avg|
x2
25.5
0.2
4.2
x2
16.7
0.2
0.9
x3
18.3
0.2
3.0
x3
22.1
0.2
4.5
xi,avg
21.3
0.2
xi,avg
17.6
0.2
R
219.3
5.2
R
250.6
2.4
35o
x0
Distance
±
45o
x0
Distance
±
348
1.5
398
1.5
x1
8.5
0.2
1.9
x1
1.7
0.2
2
x2
10.8
0.2
0.4
x2
4.5
0.2
0.8
1.5
1.2
|xi-xi, avg|
x3
11.9
0.2
x3
4.9
0.2
xi,avg
10.4
0.2
xi,avg
3.7
0.2
R
358.4
3
R
401.7
3.5
|xi-xi, avg|
49
#7 Projectile Motion
Objectives
The main objective of this experiment is to compare the actual motion
of a projectile with the motion predicted by the kinematic equations
Theory
1. Projectile launched horizontally
If a projectile is launched horizontally with an initial speed vo, the
launch velocity can be determined from the launch-to-impact horizontal
and vertical displacements of the projectile. Assuming no horizontal ac-
celeration after launch the horizontal displacement is given by
Ax = v,At
(7.1)
where At is the time of flight.
Initially, the vertical component of velocity of the projectile is zero,
but the projectile is subject to gravitational acceleration. Hence the mo-
tion in the vertical direction is that of a freely falling body with zero ini-
tial vertical velocity. The vertical displacement is therefore
1
Ay =
58(At)?
)
(7.2)
where g = 9.8 m/s
Eliminating t from Eqs. 7.1 and 7.2 yields
Vo = 4x
V2 (Ay)
(7.3)
where dy is a negative number.
2. Projectile launched at an angle relative to the horizontal
If a projectile is launched upward with an initial speed v, at an angle
0, to the horizontal, the horizontal distance traveled in a time At is
Ax = Vox At = (v, cos 0.) At
(7.4)
50
The vertical motion of the projectile will momentarily stop when
V, = Voy - gat = v, sin 0.- gát = 0)
or when
Vo sino
(
8
If the projectile lands at the same level it was launched from, the as-
cent time will equal the descent time -- the time of flight At is given by
2At, or
2vo sino
Atf
(7.5)
g
The range, R, is defined as the horizontal distance traveled by the
projectile in returning to the height from which it was launched. Thus, in
Eq. 7.4, 4x = R when Ay
R when 4y = 0. Note that when 4x = R, At = Atf. If Atf
from Eq. 7.5 is substituted for At in Eq. 7.4 and R substituted for Ax, we
find, after using a trigonometric identity,
v
Ax = R = sin 20.
8
(7.6)
It is important to remember that Eqs. 7.5 and 7.6 are valid only
when the impact point is at the same height as the launch point!
Apparatus
Make sure that you have each of the following on your lab bench:
- Spring gun
Catcher box
Meter stick
Adjustable inclined plane
Carbon paper
1.
Experimental Procedure
Place the spring gun near the edge of the lab table. Fire a few shots to
determine the approximate location where the ball lands on the floor.
Adjust the shelf in the box to its lowest position. Tape both carbon
paper and white paper to the bottom of the box and place it so that the
ball will strike the box's shelf near the center.
$1
2.
3.
4.
Launch the ball three times. Be sure that neither the apparatus nor the
box moves during the three trials.
Do not move either the apparatus or the box. With the ball in the gun
in the unloaded position carefully measure and record the vertical dis-
placement from the bottom of the ball to the floor (yo) as well as the
distance from top of the box's shelf to floor (ytb). (Subtracting these
two values gives the vertical displacement of the ball from launch to
impact.) Measure and record the horizontal displacement from the tip
of the gun in the unloaded position to the front of the box. Finally,
measure and record the distance of each of the three spots from the
front of the box.
Place the spring gun on the inclined plane on the floor. Set the angle
of the plane to 15º. Set the box's shelf at the same vertical height as
the bottom of the ball when it is in the gun in the unloaded position.
Place the box containing both carbon paper and white paper so that the
ball will strike the bottom of the box near the center.
Launch the ball three times. Be sure that neither the apparatus nor the
box moves during the three trials. Measure and record the horizontal
displacement from the tip of the gun in the unloaded position to the
front of the box. Measure and record the distance of each of the three
spots from the edge of the box.
Repeat steps 4 and 5 for three additional angular settings of the in-
clined plane: 25°, 35°, and 45°.
5.
6.
Analysis of Data
The kinematic equations can be used to analyze and make predic-
tions about the motion of an object that undergoes a constant accelera-
tion. One of the goals of this activity is to assess the validity of using
the kinematic equations to analyze the motion of the projectile used in
the activity.
We will assume that the launch speed of the ball is the same
whether the ball is launched horizontally or at an angle to the hori-
zontal.
Equation 7.3 can be used to determine the launch speed of the ball
when it is launched horizontally from some initial height. We will do
this through straight calculation using our data.
Equation 7.6 can be used to determine the launch speed of the ball
when it is launched at an initial angle from the horizontal. We will do
this through graphical analysis using our data.
52
1. Determining the launch speed of the projectile - Equation 7.3
Find the average total horizontal distance the projectile traveled after
being launched horizontally, along with its uncertainty. (Remember, you
need to determine both the average measurement uncertainty and the ac-
cidental uncertainty to determine the uncertainty of the distance.) Use
this result to determine the initial launch speed of the projectile, along
with its uncertainty.
2. Determining the launch speed of the projectile - Graphical
Analysis and Equation 7.6
For each launch angle, determine the range R of the projectile along
with its uncertainty. Use EXCEL to plot R versus sin 20., and determine
the equation to the best-fit line. Equation 7.6 predicts both the slope
and y-intercept of this line: slope = vz/g, y-intercept 0. Use the slope
of the best-fit line, along with the slope's uncertainty, to determine the
launch speed of the projectile.
-
3. Comparison of results
Compare the values of vo obtained above, using both a percent differ-
ence comparison and a comparison by evaluating agreement within their
uncertainties.
Conclusion
Don't forget to include a table summarizing your final results.
When discussing agreement between experimental and expected val-
ues, be sure to explicitly discuss the overlap of the range of values
indicated by their uncertainties. Percentage difference does not
necessarily indicate agreement between two numbers.
Your conclusion should address the validity of using the kinematic equa-
tions in analyzing the motion of the ball used in this activity, with your
arguments reinforced by the results obtained for the launch speed of the
ball.
Questions
Assuming that the motion of the projectile in this activity can be ana-
lyzed using the kinematic equations, predict the projectile's range for
a launch angle of 65°, and for a launch angle of 75°. Compare these
results to the experimental values obtained for a launch angle of 25°
and 15°, respectively, and discuss the results of your comparison.
2. What is the range for 0. = 90°? Does your answer depend on the use
of the kinematic equations? Explain.
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