Report on Projectile Motion

Projectile Motion

PHYS111
Formal Report 2

University of Canterbury

Campbell Moulder

Abstract
The force of gravity is said to be a constant of 9.81 ms-2 (3). This can be proved by measuring the projectile motion of a bouncy ball and plotting a ∆Vertical Velocity vs. Time graph, the gradient of which should equal the constant force (acceleration due to) of gravity. Our gradient value of 10.26±0.49 ms-2 is consistent with the actual value of 9.81 ms-2.
Introduction
A projectile is an object that has been launched into the air. Once a projectile has been launched, the only forces acting are:
Air friction (this is considered negligible in our experiment)
Lift force, if the object is behaving like a wing (this is also negligible as our object is a ball)
Gravity, the weight force which acts downwards (this is the value we will be calculating in our experiment)
In our experiment we will measure the projectile motion of a bouncy ball using the computer programme Motion Tracker. The results of this experiment will allow us to plot a ∆Vertical Velocity vs. Time graph. The only force that is affecting the ball that we are taking into account is the force due to gravity; therefore the gradient of this graph will give us the value of the force or acceleration due to gravity. The horizontal component is negligible because once the ball has left our hand the only force acting is downward, the horizontal force remains constant. The objective of our lab experiment is to either confirm or deny the hypothesis that the value of gravity is a constant and is equal to 9.8ms-2.
Materials and Methods
1 Computer with webcam and projectile motion analysis program installed
1 Lightweight Bouncy Ball
1 Carpenters Square with wooden support
1 Metre Ruler
1 Clamp Stand
We set up the apparatus according to the PHYS111 lab manual: We stood the Carpenters Square up on the bench and attached the camera to the clamp stand at a height so the Carpenters Square ran parallel with the x and y axis of the camera. We then bounced the ball through the path of the camera and recorded its motion. This proved to be difficult at times because the ball had to travel along the x-axis of our carpenters square in order for us to get an accurate example of projectile motion. Also the ball had to bounce within the boundaries of the camera so that the full motion of the ball could be recorded. Using the motion tracker program we then marked out centre of the ball frame by frame as it travelled across the screen. We took the x and y values of the plotted points and this gave us our projectile motion graph. We then plotted a Velocity VS Time graph and took the gradient; this gave us our value of gravity.
Results and Analysis (1)
Table 1 Time (s) x and y axis (cm) results from Motion Tracker programme t (s) | x (cm) | y (cm) | 0 | 0 | 6.1 | 0.031 | 3.4 | 11.5 | 0.063 | 7 | 15.9 | 0.094 | 10.5 | 19.6 | 0.125 | 14.4 | 22.7 | 0.156 | 18.2 | 24.7 | 0.203 | 22 | 25.5 | 0.25 | 25.8 | 25.2 | 0.281 | 29.9 | 24 | 0.313 | 34 | 21.4 | 0.344 | 38.1 | 17.6 | 0.375 | 42.2 | 12.4 | 0.406 | 46.1 | 6.1 |

Figure 1 The Projectile Motion of a bounce ball We then manipulated the data from Table 1 to calculate ∆t (s), ∆y (cm), Vy (cm/s) and Vy (m/s). To calculate ∆t we took the difference between two consecutive time values e.g. 0.32-0.31=0.31, therefore the change in time for the given two will be 0.031s. To calculate the ∆y we used the same process as for the ∆t. We then needed to calculate the Velocity of the ball in the y direction. To calculate the velocity of an object the formula v=d/t can be used. From the results of our experiment we knew the distance the ball travels and the time it takes. This allowed us to calculate the vertical velocity of the ball. Table 2 ∆t, ∆y and the velocity of the ball in the vertical direction ∆t (s) | ∆y (cm) | Vy (cm/s) | Vy (m/s) | 0 | 0 | 0 | 0 | 0.031 | 5.4 | 174.2 | 1.742 | 0.032 | 4.4 | 137.5 | 1.375 | 0.031 | 3.7 | 119.4 | 1.194 | 0.031 | 3.1 | 100 | 1 | 0.031 | 2 | 64.5 | 0.645 | 0.047 | 0.8 | 17 | 0.17 | 0.047 | -0.3 | -6.4 | -0.064 | 0.031 | -1.2 | -38.7 | -0.387 | 0.032 | -2.6 | -81.3 | -0.813 | 0.031 | -3.8 | -122.6 | -1.226 | 0.031 | -5.2 | -167.7 | -1.677 | 0.031 | -6.3 | -203.2 | -2.032 |

Figure 2 the vertical velocity of a ball moving with projectile motion
Half of our results were negative. This is due to the direction of the ball. Velocity is a vector and therefore has both a direction and a magnitude. This graph is only showing the vertical velocity so half of the projectile motion movement will be negative due to the ball falling.
We took the gradient of the line of best fit and this was our gravity value. The value we calculated was found to be 10.26 ms-2. We then took a line of worst fit and calculated the difference between the two gradients which gave us 0.49. This number was our uncertainty. We calculated the value of gravity to be 10.26±0.49 ms-2. This value agrees with the value of given in our PHYS111 lab workbook of 9.81 ms-2.
My gradient line uncertainty is about 5% (10.26±0.49 = 10.26±5%). Based on this I have chosen my error bars on the projectiles vertical velocity to be ±5% of each velocity value. (2)
Discussion
Our experiment tested the hypothesis that gravity is a constant force. We found this to be accurate. We calculated the value of the acceleration due to gravity to be 10.26±0.49 ms-2. The true value of this acceleration is 9.81 ms-2 (3); this is within our values uncertainty showing that our experiment was a success. This was an optimal result and complied with what we expected. The basic data produced by the programme Motion Tracker made a perfect projectile motion graph (see Figure 1). This raw data that we collected made it obvious that we had carried out the first part of the experiment successfully. After we had manipulated the data to calculate the vertical velocity during each time period we produced a relatively linear graph (see Figure 2). This was as expected as we needed a linear graph to calculate the value of gravity.
We know that the linear graph gives us the relationship v=(gradient).t
And we know that a=∆v/t
Rearranging the formula we get ∆v=a.t
Therefore the gradient of the graph should give us the acceleration experienced by the ball (i.e. gravity g=9.81 ms-2)
Conclusion
Our experiment showed that the acceleration experienced by the ball is due to the force of gravity i.e. our calculated acceleration was 10.26±0.49 ms-2 and acceleration due to the force of gravity is 9.81 ms-2. 9.81 ms-2 falls within our calculated range of 9.77 ms-2 and 10.75 ms-2.
There was limited uncertainty in our experiment as the computer calculated our values for us. This made the possibility of human error very small.
Our experiment upheld the hypothesis that gravity is a constant force and the acceleration due to this force can be calculated from the gradient of a Vertical Velocity VS Time graph.
References
1 Table of Results
All results were taken from my PHYS111 Lab Book
2 Calculation of vertical velocity uncertainties
I was unsure as to how to draw the error bars on the vertical velocity (as the uncertainty in time was stated to be negligible). I decided to draw the error bars as a 5% uncertainty on each vertical velocity measurement. I chose 5% because the uncertainty on my gradient was 5%.
3 The value of gravity is given in several texts. I used: Advanced Senior Physics Edited by N. F. Barber and R. J. Osborne Heinemann Educational Books (NZ) Ltd. Published 1979

References: 1 Table of Results All results were taken from my PHYS111 Lab Book 2 Calculation of vertical velocity uncertainties I was unsure as to how to draw the error bars on the vertical velocity (as the uncertainty in time was stated to be negligible). I decided to draw the error bars as a 5% uncertainty on each vertical velocity measurement. I chose 5% because the uncertainty on my gradient was 5%. 3 The value of gravity is given in several texts. I used: Advanced Senior Physics Edited by N. F. Barber and R. J. Osborne Heinemann Educational Books (NZ) Ltd. Published 1979