Projectile Motion: Principles And Experiment

Aim

A projectile is an object that has the ability to move under gravitational force. The movement of a projectile is normally under constant speed in the horizontal direction and a constant acceleration downwards in the vertical direction. Such acceleration is due to the gravity of the earth. Projectile motion hence defines a motion of a projectile that has been launched into the air in the presence of an applied force at the start and thereafter the projectile will be moving along a curved path as it experiences gravitational force [1]. Projectile motion finds its application in real life in numerous fields which range from applications in the military and sporting to satellite motion applications. Among the examples of such applications include path of the ball in footballs and cricket in sporting and projectile of mortars, tanks shells and bullets in military applications.

This laboratory experiment aimed at evaluating the properties or principles of projectile motion for a simple ball and ramp configuration where the ball is released from a certain height and launched at a present angle. The design of this laboratory is don is such a way that it fosters a comprehension of the principles of rotational, potential and kinetic energies besides and understanding of the vertical alongside horizontal components of acceleration, velocity and position. 

This laboratory experiment was based on three main objectives:

• To calculate the initial velocity of a ball that has been shot horizontally using a projectile launcher
• To find the angle of projection that produces the maximum rate
• To identify the average projection rate of the ball using an angle set

Projectile motion is one of the examples of motion that have constant acceleration when the resistance of air is ignored. An object turns to be a projectile at the moment it is released and is under the influence of gravity.

The x- and y-components of the motion of a projectile are independent of each other and only connected by the time of flight, t [2]. The position as well as the velocity of a trajectory of a projectile can be determined using some of the kinematics equations. The projectile can be either in a horizontal or vertical position. The equation  is used in the determination of the postion when the projectile is undergoing horizontal motion since gravity does not act on the x component. In the vertical motion, the formula  is used in the calacualtion of the position of the projectile [3].

Owing to the fact that acceleration is applied in the vertical direction, the velocity in a horizontal direction is normally constant. For this reason, the velocity id a vertical direction is determined using the linear expression. Upon the object obtaining its maximum height, the vertical velocity becomes zero while the acceleration is still that os the gravity of the earth a=-9.8 m/s².

Still, when a projectile is released or fired from a certain angle, trigonometry is used in the calculation of the horizontal and vertical velocities.

When the angle is flat, the projectile will move faster in the horizontal direction as compared to how it moves on a steep angle [4]. The total velocity of the projectile is determined through summing up the velocity in the y component and the velocity in the x components through the use of Pythagoras theorem

Objectives

Just to illustrate the application of the Pythagoras theorem, when a ball is thrown from a certain angle, the motion of the ball would be described as a parabola. There would be two forces acting on the ball at various times: the contact force and the gravitational force but this situation changes as soon as the ball gets to the air in which on gravitational force will be acting upon it.

The equation  descirbes the postion of the ball in the horizontal direction where a=0 and hence  [4]

Below is a theoretical graph of the position against time of the ball when in the vertical direction which would be a parabola which opens in the downward direction. The ball position increases to the point when the maximum height is reached upon which the position would then decrease and be restored to its original position. In such a case, the acceleration which is due to gravity would have components only in the y, hence a=g=-9.8 m/s². The position of the ball would thus be given by the equation  [5]

The velocity against time graph in the horizontal direction tends to be a straight line since the ball is not undergoing any acceleration and thus has a constant velocity. Velocity is a derivative of position and the horizontal direction position was given by the equation  hence  where C is a constant [5].

The graph of the velocity against the time in the vertical direction turns out to be inclined line and since the velocity is a derivative of position and the ball position is in the vertical direction, the equation   is used in determining the position [6]. The derivative of the above preceding quadratic equation results in a linear expression, hence.

1. About 50 cm of black tape was tapped down from the ramp base along the horizontal axis as can be observed in Appendix C.
2. The Logitech software was loaded and the camera ensured to be in position as demonstrated in Appendix C
3. One of the ball sizes, launch angle and height of releasing ball was selected. (The angle chosen was ranging between 8?and 25?) and their parameters recorded in Appendix B. the ball bearing was released using the camera software from the chosen height and the path of the ball recorded [7].
4. The video was imported into Tracker and analysis done on it as per the instructions in Appendix C so as to get the measurements of the (x, y) positions as well as the velocity at point B. The obtained data was copied into an Excel document[8].

Note: The recorded videos were to be found in the default folder.

5. A graph of the trajectory path of the ball with regards to the (x, y) position was plotted in Excel
6. A theoretical analysis was performed (Appendix A) to find the equation of the ball describes the position of the ball correctly with respect to (x, y) position[8].
7. On the same graph that is generated in the preceding steps, a trajectory path of the ball with regard to the (x, y) position was plotted:

• The theoretical path of the ball using the measured velocity
• The theoretical path of the ball using the theoretical velocity

Note: Only the data points were used during the plotting of the experimental data while a continuous line was used for the theoretical data [4].

8. The steps 3-7 were repeated using different ball sizes, release angle and height until when a total of three tests had been carried out

Test for Mass A

t	x	y	v
0.00E+00	-4.29E-02	1.59E-01
3.20E-02	1.50E-02	1.65E-01	1.72E+00
6.40E-02	6.65E-02	1.69E-01	1.71E+00
9.60E-02	1.24E-01	1.57E-01	1.99E+00
1.28E-01	1.87E-01	1.26E-01	2.20E+00
1.60E-01	2.45E-01	8.44E-02	2.32E+00
1.92E-01	3.00E-01	3.00E-02

Table 1: Test Results for Mass A

Test for Mass B

T	x	y	v
0.00E+00	-5.98E-02	1.69E-01
3.20E-02	-1.90E-02	1.80E-01	1.29E+00
6.40E-02	2.04E-02	1.87E-01	1.19E+00
9.60E-02	5.71E-02	1.84E-01	1.30E+00
1.28E-01	1.01E-01	1.64E-01	1.60E+00
1.60E-01	1.45E-01	1.32E-01	1.80E+00
1.92E-01	1.84E-01	8.50E-02	2.08E+00
2.24E-01	2.22E-01	2.38E-02

Table 2: Test Results for Mass B 

T	x	y	v
0.00E+00	-3.19E-02	2.05E-01
3.20E-02	-5.32E-03	2.14E-01	7.95E-01
6.40E-02	1.75E-02	2.17E-01	8.50E-01
9.60E-02	4.87E-02	2.08E-01	1.04E+00
1.28E-01	7.45E-02	1.82E-01	1.38E+00
1.60E-01	1.09E-01	1.43E-01	1.81E+00
1.92E-01	1.41E-01	8.74E-02	2.11E+00
2.24E-01	1.70E-01	2.28E-02

 Table 3: Test Results for Mass C 

All the tests A, B and C were done from different ball sizes, release heights and release angles. Each of the masses A, B and C had the same launch angle at 0?. However the measurement of Mass A shot the ball at the longest range and Mass B shot the ball at a medium range. Mass C had the shortest range of shot of the ball. As can be observed in the results tables the maximum flight time for Mass A was 1.92E-01, 2.24E-01 for Mass B and 2.24E-01 for Mass C, providing maximum flight times that are equal for masses B and C while nearly equal masses for Mass A as compared to the other two masses [8].

The initial velocities for the three masses are however significantly different where Mass A had the highest initial velocity at 1.72E+00 m/s and Mass C having the least initial velocity at 7.95E-01 m/s. the higher velocity for Mass A illustrated the higher range of measurement obtained for the mass which did not have a relatively significant change in the time of flight as compared to Mass C [3]. To determine the value of y_max by adopting the equation  for the measurement of the time taken by the ball to get to y_max and then using the time to solve the equation .

One of the limitations of this experiment pertains taking the measurements of flight range which is the horizontal distance from the point of releases to the point of landing on the timing pad which might have a slightly percentage errors owing to human error for accurate measurement [9].

In conclusion, a projectile may be classified into motion in the vertical direction and horizontal direction. The vertical motion direction is determined using the equation  . The objectives of the experiment were met with the experimental graphs showing a significant correlation with the theoretical graphs.

References

[1]	David Loyd, Physics Laboratory Manual, David Loyd, Ed. New York: Cengage Learning, 2013.
[2]	Jerry D. Wilson, Physics Laboratory Experiments, 8th ed., Jerry D. Wilson, Ed. London: Cengage Learning, 2014.
[3]	Vishal Mody, High School Physics: Projectile Motion, 4th ed., Vishal Mody, Ed. London: CreateSpace Independent Publishing Platform, 2015.
[4]	Rhett Allain, Physics and Video Analysis, 6th ed., Rhett Allain, Ed. New York: Morgan & Claypool Publishers, 2016.
[5]	John Wiley & Sons, Fundamentals of Physics, 9th Edition, Halliday-Resnick-Jearl Walker, 2011: Fundamentals of Physics, 5th ed., John Wiley & Sons, Ed. New York: Bukupedia, 2011.
[6]	Colin White, Projectile Dynamics in Sport: Principles and Applications, 5th ed., Colin White, Ed. Berlin: Routledge, 2010.
[7]	Joseph Gallant, Doing Physics with Scientific Notebook: A Problem Solving Approach, 6th ed., Joseph Gallant, Ed. London: John Wiley & Sons, 2012.
[8]	John Matolyak, Essential Physics, 6th ed., John Matolyak, Ed. Moscow, Russia: CRC Press, 2013.
[9]	Narciso Garcia, Physics for Computer Science Students: With Emphasis on Atomic and Semiconductor Physics, 5th ed., Narciso Garcia, Ed. New York: Springer Science & Business Media, 2012.
[10]	Kerry Kuehn, A Student's Guide Through the Great Physics Texts: Volume II: Space, Time and Motion, 4th ed., Kerry Kuehn, Ed. Toronto: Springer, 2014.