Experimental General Physics for Engineering I
PHYS 192 – Fall 2021
Experiment Title: Projectile
Student Name: Dhabya mana AlQahtani
Student ID: 201507830
Lab list number: L62
Lab list number:
Experiment Date: 21/9/2021
Due Date:
Instructor: Mohammed Vallikkaparambil
Submitted Date: 28/9/2021
Percentage Similarities:
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Points Earned:
Projectile
Table of Contents
Abstract: ..................................................................................................... 1
Equations:................................................................................................... 1
Equipment: ................................................................................................. 2
Implemented Procedure:............................................................................. 2
Data: ........................................................................................................... 3
Analysis: ..................................................................................................... 3
Graphs: ....................................................................................................... 4
Results:....................................................................................................... 5
Sample Calculations: .................................................................................. 5
Discussion and Conclusion: ........................................................................ 7
References: ................................................................................................ 8
Projectile
Abstract:
An object launched into the air at the surface of the earth is known in physics as a
projectile. Its motion is called a ballistic motion. Once the projectile is launched, it
moves under the influence of the earth’s gravitational force and of forces exerted
by the air. Here we will neglect the effects due to the air. During the launch process
itself, other forces are exerted on the object, usually for a short period of time,
which result in a certain initial velocity, initial height and a certain launch angle.
With the knowledge of those initial conditions, it is possible to calculate the
complete trajectory of the projectile. Among other quantities, one can compute its
range (horizontal distance travelled), its maximum height, and its time of flight. In
this experiment, we will focus on the range.
Equations:
The constant acceleration is then a0  g , and the y-components of the motion
are given by:
y(t)  y0  voyt 
1 2
gt
2
Eq. (1)
v y (t)  voy  gt
Eq. (2)
a y (t)  g
Eq. (3)
y0  y(t  0) and v0 y  v y (t  0) are the vertical components of the position and of
the velocity of the projectile at time zero. Here, the time at which the projectile is
released is taken as time zero.
The horizontal component of the motion is given by:
Experimental General Physics for Engineering I, PHYS 192 – Fall 2020
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Projectile
x(t )  x0  voxt
Eq. (4)
vx (t)  vox
Eq. (5)
ax (t)  0
Eq. (6)
x0  x(t  0) and v0 x  vx (t  0) are the horizontal components of the position and
of the velocity of the projectile at time zero.
The launch angle  (measured from the horizontal) and the initial velocity v0
determine the values of v0 x and v0 y :
v0 x  v0 cos( )
Eq. (7)
v0 y  v0 sin( )
Eq. (8)
In this experiment, we will measure the initial velocity of a projectile ball v0 , the
launch angle  and the horizontal distance travelled between by the ball before it
hits the ground, known as the range R  x(t f )  xo , where t f is the time at which
the ball hits the ground and therefore terminates its projectile motion.
Considering the case where the initial and final heights are the same, the equations
lead to:
v sin(2 )
R 0
g
2
Eq. (9)
Equipment:
-Power supply.
-Ballistic launcher unit with velocity meter.
-Meter rod.
-Carbon paper.
-Speed measuring tool.
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Projectile
Figure 1: Projectile launcher experimental setup
Implemented Procedure:
1. Listen to and follow the instructions of the instructor very carefully. Do
not place your fingers in the launching area !
2. The experimental setup is as shown in Figure 3. 1
3. Make sure the ballistic unit is aligned with the bench and that the bench is
covered with carbon paper. There should be no space between the unit and
the bench.
4. Adjust the angle according to the value given in Table 3.1.
5. Record the angle and its uncertainty.
6. Load the spring in the first position.
7. Release the spring, the ball will start its projectile motion.
8. Record the initial velocity, as given by the velocity meter.
9. Find the mark made by the ball on the carbon paper.
10. Measure the horizontal distance travelled by the ball (range).
11. Repeat the steps 6-10 two more time.
12. Change the launch angle according to Table 1.and repeat the steps .
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Data:
Launch
Uncertainty
angle 
on launch Trial Trial Trial
()
angle
Initial velocity
#1
Range
Trial
Trial
Trial
#2
#3
#2
#3
#1
35
0.5
2.70 2.68
2.68
0.761
0.763 0.764
45
0.5
2.68 2.68
2.69
0.809
0.807 0.807
50
0.5
2.73 2.72
2.69
0.801
0.791 0.803
60
0.5
2.70 2.63
2.67
0.686
0.689 0.691
55
0.5
2.55 2.36
2.56
0.671
0.667 0.676
Table 3. 1: Data table
Analysis:
1. Compute the average initial velocity v0 av from the 3 trials at each angle.
2. Calculate its uncertainty using the standard deviation of the mean method.
3. Calculate the average initial velocity of all trials and its uncertainty.
4. Compute the average range Rav from the 3 trials at each angle.
5. Calculate its uncertainty using the standard deviation of the mean method.
6. Calculate the value of sin(2 ) for each angle, then its uncertainty using the error
propagation method.
7. Plot a graph sin(2 ) (y-axis) Vs. Rav (x-axis).
8. Add error bars to your graph.
9. Fit your data; get the slope, the y-intercept, and the fit quality parameter R 2 .
10. Find the uncertainties in the slope and intercept using the LINEST function in
Excel.
11. From the value of the slope and using Eq. (9) calculate the corresponding value
of v 0 and its uncertainty (use g =9.8 m/s2).
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Average initial
Launch
angle 
sin(2 )
u(sin(2 ))
velocity v0 av
Average
Range
u(v0 av )
()
u(Rav )
Rav
35
0.9397
0.009
2.6867
0.01
0.7626
0.1
45
1
0.006
2.6833
0.01
0.8076
0.1
50
0.9848
0.00
2.7133
0.01
0.7983
0.1
60
0.866
0.009
2.6733
0.01
0.6887
0.5
55
0.9396
0.006
2.49
0.01
0.759
0.1
Table 2. 2: Data Analysis table
Graphs:
‫عنوان المخطط‬
y = 0.8943x - 0.0828
R² = 0.9941
0.82
0.8
‫عنوان المحور‬
0.78
0.76
0.74
0.72
0.7
0.68
0.84
0.86
0.88
0.9
0.92
0.94
0.96
0.98
1
1.02
‫عنوان المحور‬
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Results:
From the graph of average range versus sin(20), the slop is
∆𝑅
= 82.65 𝑐𝑚
∆ sin(2𝜃)
We use the unit cm for the range in order to keep accuracy of the measurement .
𝑠𝑙𝑜𝑝𝑒 = 0.8265 𝑐𝑚
Comparing with
𝑠𝑙𝑜𝑝𝑒 =
𝑣𝑜2
𝑔
The anitial velocity is
𝑣𝑜 = √9.8 × 0.8265 = 2.85 𝑚⁄𝑠
Then the accuracy of the result is
𝑥2 = ∑
(𝑣1 − 𝑣0 )2
𝑣1
With doing this equation we get finally
𝑥 2 = 0.058
Sample Calculations:
1-Calculation of average initial velocity
𝑣𝑎𝑣 =
𝑣1 − 𝑣2
2
For angle = 35
𝑣𝑎𝑣 =
2.69 − 2.71
= 2.70 𝑚⁄𝑠
2
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2- Calculation of uncertainty in average initial velocity.
𝜎𝑣 = √
∑𝑛𝑖=1(𝑣1 − 𝑣0 )2
𝑛−1
For angle =35
𝜎𝑣 = √
∑21(2.69 − 2.70)2 + (2.71 − 2.70)
= 0.014142136
2−1
𝜎𝑎𝑣 =
𝜎𝑣
√𝑛
=
0.014142136
√2
= 0.01
3- The average range for angle =35 is
𝑅𝑎𝑣 =
𝑅1 + 𝑅2 68.5 + 69.2
=
= 68.85 𝑐𝑚
2
2
4-The uncertainty in average range is
𝜎𝑅 = √
∑21(68.5 − 68.85)2 + (69.2 − 68.85)2
= 0.494974747
2−1
𝜎𝑅𝑎𝑣 =
𝜎𝑅
√𝑛
=
0494974747
√2
= 0.35
5-Calculation of uncertainty in sin(2𝜃)
∆sin(2𝜃 ) =
sin(2[𝜃 + ∆𝜃 ]) − sin(2[𝜃 − ∆𝜃 ]) 0.8746 − 0.8572
=
= 0.0087
2
2
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Discussion and Conclusion:
In the experiment the ballistic motion of the projectile is investigated through the
Detailed study of the dependence of the range of the initial velocity and launching
angle.
The data in table 1 tells a lot about the projectile motion. The trend of range value
Is completlely consistent with the formulation of equation (9).
The maximum measured range is occurred at launching angle 45,which is
predicted from the theoretical analysis of the projectile motion.
The gragh average range versus sin( 2𝜃) shows a regular increase in the range
with the value of sin(2𝜃) . in final result all quantities are considerdin SI unite.
The estimated slope of the least square fit of the data is 82.65 cm , which leads to
an estimated initial velocity of 2.85 m/s ,in order to estimatethe accuracy of the
estimated initial velocity, Chi-square test is performed and value of Chi-square is
0.058 . this value is reatively high because rhe errors.
The results presented in this report isconsistent with the theoretical formulation of
the projectile motion as the effect of the air resistance is negligible because of the
small size of the projectile and the short time of flight. For the projectile large size
and longer time of flight.
The equation and analysis presented in this report gives exact results for the
projectile motion in absence of air resistance, and good approximation for the
motion with air resistance.
Finally, the time of the experiment is fulfilled by the investigation of the
correspondence between the inferred value and the true value of initial velocity,
and the dependence of the range on initial conditions.
The experience is really inspiring, as it ispossible to design many experiment on
the projectile motion.
References:
1. PHYWE series of publications • Laboratory Experiments • Physics • P2131100
© PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
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