Experiment 4: Projectile Motion
Introduction:
Projectile motion is a special type of motion involves two components of motion – horizontal and vertical. But motions in two directions are independent of each other. Flight of ball is an example of projectile motion. This two dimensional motion can be determined as two linear motions in two mutually perpendicular directions –one along horizontal direction and the other one along vertical direction. Projectile motion can be analyzed with the help of equations of motion. Vector equations and vector rules are generally used as the motion occurs in two dimensions. Gravity force has primary importance in projectile motion. Effect of other forces such as air resistance is negligible. Vertical component of velocity of a body in projectile motion increases linearly as the acceleration due to gravity is constant and horizontal component of velocity remains unchanged throughout the motion. As there is no acceleration in horizontal direction the horizontal component of velocity is constant. Purpose of this experiment was to have knowledge about projectile motion like mathematical expression of projectile motion, relation between horizontal projectile’s range with time of flight etc.
Theory:
In this experiment a ball bearing was rolled down a curved track and allowed to fly off the end of the lab table. Just before the ball bearing had flied off the end of the track, the ball rolled through a photogate. After the ball bearing travelled through the photogate, the computer’s software recorded the time taken by the ball to block the beam as Δt. If the ball was centered on the infrared beam, this was the amount of time the ball travelled in a distance of one ball diameter d. So, ball left the track with its initial velocity, vox = d/ Δt.
Even though several forces acted upon the ball bearing as it rolled down the track, attention was paid to the ball’s motion only after it started to fall. As it left the end of the track, the ball had an initial velocity in the X direction and Earth’s gravity pulled the ball toward the floor with a constant force. For this reason the X component of the ball’s velocity was great and it did not have Y component of initial velocity. As gravity pulled the ball down, its Y-velocity increased. The initial X-velocity and the increasing Y velocity were always perpendicular to each other and those two influences were independent of one another. So, the ball’s motion was along the vector sum of those two velocity vectors. Assuming negligible air resistance, the ball executed uninterrupted constant velocity motion in the X direction and uninterrupted accelerated motion in Y direction.
A variable x is assigned for the range of the ball bearing. Time taken by ball to fall to the floor is t and voxis the initial velocity in X direction.
The equation for constant velocity motion is
x = xo + vox.t.. (1)
In the same time, the ball falls down a height h. The ball started falling with zero initial downward velocity. So,
h = ho + vo.t + ½ . g.t^2
As ho and voare equal to zero, from above equation following expression can be derived.
t = (2h/g)^0.5
Now, putting the value of t in equation no. 1,
x = xo + vox.(2h/g)^0.5.. (2)
Procedure:
In this experiment a curved track, a ball bearing, a table, a photogate, a ring stand base, one-meter long strip of cash register paper, a right angle clamp and a threaded rod were used. Following figure indicates the experimental setup.
At first the track was placed at the edge of the lab bench and clamped it to the tabletop. The ball was placed on the horizontal part of the track to check whether the track was level. In the next step the photogate was connected to the ring stand base utilizing a right angle clamp and a threaded rod, screwed into the photogate. The position of the photogate was then adjusted in such a way that beam of the photogate was blocked by the full diameter of the steel ball bearing as it rolled along the last few centimeters of the track. The diameter of the ball bearing was measured with a vernier caliper and this measurement was recorded. Then Logger Pro was started. Experiment menu was pulled down and Data Collection was clicked on. Mode was set to Digital Events. This changed the data table in Logger Pro to provide the gate time (labeled GT), which was Δt and it automatically calculated the speed of the object passing through the photogate. It was ensured that Gate Timing was checked with menu commands: Experiment > Setup > Sensors > Show all Interfaces >photogate icon > Gate Timing. The diameter of the ball bearing was then entered into Logger Pro by pulling down the Experiment menu, going to Setup Sensors and Show all Interfaces. In the next step the photogate icon was clicked and the Set Distance or Length command was executed. The measured diameter of the plastic ball was entered in meters. A one-meter long strip of cash register paper was taped on the floor in front of the track. A plumb bob was used to mark on the paper the location of the end of the track. Then the ball was rolled off the end of the track and it was determined how far up the track the release mechanism could be placed before the ball would miss the end of the paper.
At least 8 trials were carried out, each trial with increasing initial velocity. For each trial the ball’s speed through the photogate was recorded in a spreadsheet. Then the height of the ball “y” was measured. After this the paper was removed from the floor and the range for each trial was measured. Each range next to its corresponding speed was recorded. Two adjacent columns were arranged in the spreadsheet where the left column with the velocities and the right with the ranges. Excel was used to plot a graph of range versus velocity. Then the data was fit with a straight line using the Add Trendline command in the Chart menu. The line’s equation was displayed on the graph. Appendix E was used to generate a scientific graph with Excel.
Data Analysis:
Diameter of the ball (d) =0.0254m.
h = 0.971m +- 0.0005m.
g=9.792m/s^2
Following data were obtained from 8 trials:
Following graph was found from the data:
g =9.792 +- 0.005m/s^2
Theoretical Value =0.445m/s^2
% diff slope=3.59%
Difference = 0.45562382 – 0.4453 = 0.0103238
Sigma d = ((0.4556238 x 0.4556238)+(0.4453 x 0.4453))^0.5 = 0.637
Conclusion:
Projectile motion is vital area of study in motion of a body and by performing this experiment knowledge about projectile motion was gained. Linear motion of projected ball bearing was continuously worked upon by gravitational force resulting in change of both magnitude and direction of velocity. Study of projectile is equivalent to two independent linear motions. The experiment was successful. Though error was found while analyzing the data, percentage of error was negligible. If more care would be taken to follow experimental procedure during experiment more accurate result could be obtained. However, purpose of this experiment was met.
