PHYS-101- Section 08 Project Final
Name: Cem Gökmen
Student ID: 22301707
Department: ME
Projectile Motion on an Inclined Plane with a Spring Gun
Objective:
This experiment’s aim is to find the spring constant and the maximum height of the
motion by calculating and examining the change in the potential and kinetical energy of the
object with different compression distances, measuring the final height of the projectile
motion and mass of the object.
Theory:
A projectile is defined as any object thrown with some initial velocity, which is then
allowed to move under the action of gravity alone. First, an object is going to be shot by a
spring gun from the bottom of an inclined plane with the object’s angle with respect to the
inclined plane of different 𝜃 angles and inclined plane’s angle with respect to the ground as
15° . Then the distance 𝑑 of the object through the inclined plane will be measured and by
using the angle the height ( ℎ ) with respect to the ground will be calculated, and the mass
( 𝑚 ) of the object is going to be measured. Afterwards, by calculating the final height ( ℎ ),
the x and y component of the initial velocity of the object are going to be calculated by only
knowing the tilt angle of the motion and the inclined plane. Furthermore, with the results
we get, the spring constant, maximum height and the initial and final velocity of the object
will be calculated using the theorems below.
Research:
Although air resistance is often important, in many cases its effect can be ignored,
and we will ignore it, and we will take the gravitational acceleration ( 𝑔 ) as 9.81 𝑚/𝑠 2 in the
following experiment.
To find the initial and final velocity of the object, we need to find the final kinetic
1
energy of the object. We define the quantity 2 𝑚𝑉 2 to be the translational kinetic energy,
K, of the object (Physics for Scientists & Engineers, 5th edition, 2021, chap. 7-4):
1
𝐾 = 𝑚𝑉 2
2
The gravitational potential energy, U, at any point a vertical height y above some
reference point (the origin of the coordinate system) can be defined as (Physics for
Scientists & Engineers, 5th edition, 2021, chap. 8-2):
𝑈 = 𝑚𝑔𝑦
According to the work-energy principle, the net work Wnet done on an object is
equal to its change in kinetic energy K (Physics for Scientists & Engineers, 5th edition, 2021,
chap. 8-3):
Wnet=ΔK.
(If more than one object of our system has work done on it, then 𝑊𝑛𝑒𝑡 and ∆𝐾 can
represent the sum for all of them.) If we assume a conservative system, we can write the
work done on an object or objects by the conservative forces in terms of the change in total
potential energy between points 1 and 2 (Physics for Scientists & Engineers, 5th edition,
2021, chap. 8-3):
We combine the previous two equations, letting U be the total potential energy:
∆𝐾 + ∆𝑈 = 0
We now define a quantity E, called the total mechanical energy of our system, as the sum
of the kinetic energy plus the potential energy of the system at any moment:
𝐸 =𝐾+𝑈
Now, from these calculations we get that ∆𝑈 = −∆𝐾 . Hence, these equations will give us
the total energy of the system, which is equivalent to the initial total energy of the system
because of the conservation of energy. Later, we will use these theorems to calculate the
spring constant.
Since, we will be working on an inclined plane, to find the initial velocity, first we will find
the y component of the initial velocity, where 𝛽 and 𝛼 as the tilt angle of the object with
respect to inclined plane and the tilt angle of the inclined plane to the ground. The
displacement of height ( ℎ ) of the object as a function of time is defined as:
1
ℎ = ℎ0 + 𝑉0 sin(𝛽 + 𝛼) + 𝑔𝑡 2
2
And also, for the x direction where 𝑥 is the displacement of the object with respect to
ground and 𝑉0 as the initial velocity and 𝑡 as the time interval of the object’s motion, we can
also use:
𝑥 = 𝑉0 cos(𝛼 + 𝛽) 𝑡
From the equations above, we will find the time interval 𝑡 and the initial velocity 𝑉0. Also,
we will be able to find the final velocity of the object too. We can use the given equation to
find the maximum height that the object will reach:
ℎ𝑚𝑎𝑥 =
(𝑉0 sin(𝛽 + 𝛼))2
2𝑔
We can say that the sum of the final potential and kinetic energy is equivalent to the sum of
the initial potential and kinetic energy of the object by the Law of Energy Conservation.
Thus, we get:
𝑈𝑖 + 𝐾𝑖 = 𝑈𝑓 + 𝐾𝑓
And since, the object has no height on the y coordinate at the initial position, 𝑈𝑖 must be
equivalent to the potential energy of the spring, which is stated as:
1
𝑈𝑖 = 𝑘𝑥 2
2
Where 𝑘 as spring constant and 𝑥 as the distance between the stretched and the
unstretched position of the spring which we will measure during the experiment. We can
also find 𝑘 by sketching a graph of 𝐸 − 𝐾𝑖 versus 𝑥 2 (spring distance):
1 2 1
𝑘𝑥 = 𝑚𝑉𝑓2 + 𝑚𝑔ℎ
2
2
Thus:
1 2 1
𝑘𝑥 = 𝑚𝑉𝑖2 = 𝑚𝑔ℎ
2
2
1
1
Let 2 𝑚𝑉𝑖2 = 𝑚𝑔ℎ to be 𝐸 which is the total mechanical energy, and 2 𝑚𝑉𝑖2 to be 𝐾𝑖 which is
the initial kinetic energy of the system.
Thus, we get:
𝑘 =
Equipment:
1. A light ball
2𝐾𝑖
𝑥2
2. A watch
3. A Scientific Calculator
4. An inclined plane
5. A spring tool
6. Measurement tool
7. A mobile phone
Procedure:
Part A:
-
First, we will calculate the weight of the object with our scale.
-
Our ball is going to be stitched with needles to mark the exact final position of its
motion on the inclined plane.
-
Put the inclined plane on a straight surface and adjust the tilt angle of the inclined
plane to be 15° with respect to the ground.
-
Put the spring tool to the bottom of the inclined plane and adjust the spring’s tilt
angle with respect to the inclined plane to be different angles each time.
-
Put the ball on the spring gun and compress it, measure the distance of the
compressed position of the spring to the initial position and note it on the data table
for each.
-
Start recording the experiment with a mobile phone or a camera device.
-
Release the spring and examine the motion of the ball.
-
Mark the final position of the ball on the inclined plane (where it dropped).
-
Calculate the height of the final position and put it on the data table.
-
Show the calculations on the given part, find the initial and final potential and
kinetic energy of the object and use the results to find each variable that has been
requested for this experiment (spring constant, initial velocity, maximum height and
time interval).
-
Check if the 𝑈𝑓 = 𝑈𝑖 . If not, why? Check if the 𝑘𝑠𝑙𝑜𝑝𝑒 = 𝑘𝑟𝑒𝑎𝑙 , if not, why?
-
Find the spring constant 𝑘 by sketching a graph of 2(𝐾𝑖 ) versus 𝑥 2 (compression
distance of the spring) and computing the slope of the best line of the graph.
-
Compare the results of the theoretical and the real value of the spring constant.
Data & Results:
Table 1: The Measured Variables
𝑥 (spring)
(cm)
7.5
ℎ𝑓 (final height)
(cm)
31.5
7.5
30.9
7.5
21.7
9.7
33.3
9.7
54.4
9.7
63.4
Angle (𝜃 )
(°)
30°
45°
15°
5°
10°
15°
𝑡𝑟𝑒𝑎𝑙
(s)
0.56
Angle 𝜃
30
0.63
45
0.30
15
0.26
5
0.45
10
0.52
15
Table 2: Potential and Kinetic Energy
𝑥 2 ( spring )
(𝑐𝑚2)
56.25
56.25
56.25
94.09
94.09
94.09
% of Error (Avg)
(%)
𝑈𝑖 ( 𝐽 )
𝑈𝑓 ( 𝐽 )
𝐾𝑖 = 𝐾𝑓 ( 𝐽 )
% of Error
(%)
0.030
0.030
0.030
0
0.035
0.030
0.035
14.3
0.040
0.030
0.040
25
0.080
0.030
0.080
62.5
0.110
0.050
0.110
55
0.110
0.060
0.110
45
1st 𝑥 2 values:
13.1
2nd 𝑥 2 values:
54.1
1st 𝑥 2 values:
0
2nd 𝑥 2 values:
27.5
1st values: 13.1
2nd values:
54.1
33.6 failure.
Table 3: Velocity Components
𝑘
( N/m )
𝑉𝑜
( m/s )
Real Value of 𝑘 is calculated to be 16.9
N/m. Error %
10.8
2.51
36.0
12.4
2.70
26.7
14.2
2.90
16.0
17.0
4.16
1.00
23.4
4.70
27.8
23.4
4.70
27.8
Table 4: The Spring Constant and Potential Energy
𝑘 (Spring Constant)
( N/m )
From
Real
the
value:
slope:
12.4
16.9
𝑈𝑓𝑖𝑛𝑎𝑙
(J)
𝑈𝑖𝑛𝑖𝑡𝑖𝑎𝑙
(J)
0.03
1st
0.03
2nd
0.03
3rd
0.030
1st
0.035
2nd
0.040
3rd
0.03
4th
0.05
5th
0.05
6th
0.080
4th
0.11
5th
0.11
6th
% of Error
26%
Show your work here:
1st : 0%
2nd : 14.3%
3rd : 25%
4th : 62.5%
5th : 55%
6th: 45%
Graph 1
Conclusion:
In this experiment, I have tried to calculate the spring constant k, by using the theory of
energy conservation and some simple kinematic equations. Main aim of this experiment
was to experience how energy conservation can be used in real life in complex situations
and how useful these theorems are in terms of calculating variables unknown in a system.
First, I have built a structure of an inclined plane (15 degrees) and used a spring launcher
and a ball to initiate the projectile motion. Afterwards, I set the angle between the inclined
plane and launcher to be different angles to reach better results and achieve more data. In
the experiment I expected the initial potential energy (spring potential) to be equivalent to
the final potential energy. However, because of neglecting the air resistance and the final x
component of the velocity, final kinetical energy was not equivalent to the final potential
energy. Also, as the angle between the inclined plane differ I saw that the velocity changed.
I saw that the kinetic energy was equivalent to the initial potential energy. Thus, in the
experiment the theorem I used did not satisfy my expectations. Only in the first try, as
could be seen from the data, I calculated the potential and kinetic energy the same. I saw
that small neglections led to bigger disruptions in the equations.
The Video Link for the Project:
https://youtu.be/KlsSMJg-9pY
REFERENCES:
Douglas C. Giancoli. (June 11, 2021). Physics for Scientists & Engineers, 5th edition.
Pearson.
Michel van Biezen. (2016, September 8). Physics 3.1: Projectile motion on an incline (2 of 7)
Example 2 [Video]. YouTube. https://www.youtube.com/watch?v=5ilRFw_dt4&list=PLX2gX-ftPVXX8xJoR1kH1_u5NpaH2o655&index=2
Boğaziçi Üniversitesi. Projectile Motion. chromeextension://efaidnbmnnnibpcajpcglclefindmkaj/https://physlab.bogazici.edu.tr/sites/phys
lab.boun.edu.tr/files/physfiles/3-phyl101-projectilemotion-f23.pdf
Muğla Sıtkı Koçman Üniversitesi. Experiment 4 (M6) The Projectile Motion. chromeextension://efaidnbmnnnibpcajpcglclefindmkaj/https://muweb.mu.edu.tr/Newfiles/77/77
/Experiment%205%20%28M6%29-The%20Projectile%20Motion.pdf