Department of Mechanical and Industrial Engineering
Ben Felix (000447394)
A mathematical model for the motion of a projectile has important applications in many fields. A simple mathematical model can be used to predict the displacement of an object after it has been given an initial velocity that has an x and y component. The purpose of this lab was to learn about drag force by investigating the motion of a projectile using three different mathematical models to find the horizontal distance that the projectile covers. To facilitate the experiment tennis balls were fired from a tennis ball machine at a set angle of inclination and the distance they travelled was measured. The data obtained from the experiment showed that the drag force is very dependent on the velocity of an object. The data also showed that a complex model is needed to account for the texture and shape of an object as these things can affect the drag force.
Date Submitted: Feb. 25, 2010
Lab Section: ME 3456 - 01
Course Instructor: George Adams
Lab TA: Ryan Hennessy
A mathematical model for the motion of a projectile has important applications in many fields. A simple mathematical model can be used to predict the displacement of an object after it has been given an initial velocity that has an x and y component. The purpose of this lab is to learn about drag force by investigating the motion of a projectile using three different mathematical models to find the horizontal distance that the projectile covers. One model will ignore resistance from the air and two models will incorporate the drag force. The predictions made by each model will give insight to its accuracy. To facilitate the experiment tennis balls will be fired from a tennis ball machine at a set angle of inclination and the distance they travel will be measured. Analysis of the final data will show how useful each model is under realistic circumstances. The data will be presented in the Experimental Results section and the data will be discussed in the Discussion of Results section.
The average distance that the tennis ball travelled after five trials was 67.2 feet with a standard deviation of 1.7205. Three different models were used to estimate the distance that a tennis ball would travel under the conditions used in the experiment; a simple analytical model that excludes air resistance, a simple analytical model that includes air resistance and a more complex model using MATLAB. The simple model excluding air resistance gave the largest value the next largest was the experimental distance, the next largest came from the simple analytical model with air resistance and the MATLAB program gave the smallest value. The simple analytical model without air resistance gave 73.640’, the experimental distance was 67.2’ , the simple analytical model with air resistance gave 62.162’ and the MATLAB model yielded a value of 59.687’. The calculations for each of these values are displayed on the following pages.
Tennis Ball Measured Values
The actual distance was obtained by firing five tennis balls from a machine and measuring the distance they travelled. For the purpose of data comparison the average of this data was taken. The values are displayed below in table 1.
Tennis Ball Distance (No Drag)
The calculated value for the horizontal distance travelled by the tennis ball using the simple analytical model excluding air resistance was 73.640 feet. The trajectory plot can be seen in figure 1. A diagram showing how some values were obtained can be seen in figure 1
The calculations for the simple model excluding air resistance went as follows:
Tennis Ball Distance (With Drag)
The calculated horizontal distance travelled by the tennis ball using the simple analytical model that incorporates air resistance was 62.162 feet. The trajectory of the tennis ball from the equation:
is plotted in figure 2.
The calculations for the simple analytical model that incorporates air resistance went as follows:
Tennis Ball Distance (MATLAB)
The value generated by the MATLAB program for the horizontal distance travelled by the tennis ball was 59.687 feet. The program code can be seen in appendix A. The plot of the trajectory can be seen in figure 3.
Golf Ball Analysis
The values for the motion of a golf ball were obtained using the same equations as the values for the tennis ball. The equations will not be repeated in this section.
The distance that a golf ball would travel was evaluated by the three models used with the tennis ball. The three different calculated distances of a golf ball hit at a 50 degree angle and an initial velocity of 200 feet per second can be seen in table 2. The program code used for the golf ball MATLAB calculation can be seen in Appendix A.
The simple analytical model without drag far exceeds the other two values. The simple analytical model with drag is less than the MATLAB model; this is different from the tennis ball values where the MATLAB value was the smallest. The plot of the trajectory of the golf ball can be seen in figure 4.
Discussion of Results
The simple analytical model excluding drag gives a value that is almost two feet greater than the experimental value. It is greater than the experimental value because it does not account for drag. Because there is no resistance accounted for the ball travels further. The simple model that does account for drag and the MATLAB model give values that are less than the experimental value because the equation does not account for factors such as spin on the ball and unaccounted for external forces such as a breeze in the direction of motion. Other sources of error that could cause these discrepancies are human error in taking the measurements and the possibility that the angle of inclination of the tennis ball machine could have changed over the course of many trials. Table three can be consulted for data comparison.
The simple analytical model gives a value that is much larger than the other two values because it does not account for air resistance. Air resistance will have a greater effect in this case than in the tennis ball case because the golf ball is travelling at a higher speed. The simple analytical model does not account for the dimples in the golf ball and so it yields a distance that is less than the distance that the MATLAB program yields. The dimples cause turbulent flow rather than laminar flow of air around the ball and so resistance is reduced. A plot of the trajectory of the golf ball can be seen in figure 4. Table 4 has been provided for data comparison.
The three models used in this lab to investigate projectile motion proved to have varying levels of accuracy depending on certain factors. The simple model without drag decreased in accuracy as the initial speed of the object increased because drag force is proportional to velocity squared. For the tennis ball the most accurate model was the simple model with drag. For the golf ball the value from the simple model was less than the complex model because the simple model does not account for the dimples in the golf ball that decrease drag force. The increase in drag force can be seen in the variance of the values between the MATLAB model and the simple model without drag for the golf ball compared with the tennis ball; the difference for the tennis ball, with an initial horizontal of 128.560ft/s was 677.822 feet whereas the difference for the tennis ball, with an initial horizontal velocity of 59.422ft/s was 13.953 feet. This shows that the drag force is very dependent on velocity. It can also be seen that manipulation of the shape of an object can affect the drag force.