Experiment 4 Projectile Motion by Eugenio Panero PHY 2091-06 experiment performed: Sep 13, 2006 report submitted: Sep 20, 2006 Lab Partner: Chris Hubacek Instructor: Klaus Dehmelt Introduction The purpose of this experiment is to explore the mathematics behind projectile motion. A ball with known, different initial velocities will be projected from a known height. Measurements of the travelled distance will be taken, and conclusions correlating experimental results with known formulas will be drawn. Data (See attached page) Data Analysis The distance travelled by an object with a non-constant velocity is given by the formula: − 1 𝑥 =𝑉 2ℎ 𝑔 0𝑥 Where 𝑉 is the initial velocity, ℎ is the height, 𝑔 is acceleration due to gravity, 0𝑥 and 𝑥 is the distance travelled. For the purpose of this experiment, the initial velocity is not given by 0, meaning the object at rest, but as the velocity recorded as soon as the object begins its projectile motion. A hypothetical ball that has an initial velocity of 0.25 m sec-1 and falls from a height of 0.45 m will travel the distance given by: 𝑥 = 0.25 2 ∙ 0.45 ∙ 9.81 = 0.743 𝑚 Discussion Diameter of the ball Initial height of the ball Trend line y-intercept Trend line x-coefficient Experimental value 2.9 cm ± 0.1 cm 100 cm ± 0.1 cm -1.082 0.365 Standard error 0.398 0.003 Table 1: Summary of Results Table 1 summarizes the experimental values obtained in the experiment. To obtain a better understanding of experimental error, a theoretical data set should be computed. This is done by applying the formula that has been presented in the previous section. Trial # 1 2 3 4 5 6 7 8 9 Velocity (cm / sec) Theoretical Distance (cm) 181.250 161.111 152.632 138.095 120.833 87.879 72.500 54.717 38.667 Experimental Distance (cm) 81.8 72.7 68.9 62.4 54.6 39.7 32.7 24.7 17.5 Table 2: Theoretical and Experimental Distances Percentage Error 64.7 58.8 54.1 49.7 42.4 30.9 25.4 18.7 12.7 20.94 19.17 21.50 20.29 22.29 22.13 22.41 24.31 27.26 A comparison between the theoretical and the experimental differences emphasizes an important percent error: (20.03% average). Such an error is beyond the limits of an experimental margin. The percent error will be visible also when graphically representing the best fit line of the experimental and the computational values. Comparing Theoretical and Experimental Projectile Motion Distance (cm) 90.0 80.0 70.0 60.0 50.0 Theoretical Distance 40.0 y = 0.4515x 30.0 20.0 Experimental Distance 10.0 y = 0.3649x - 1.0819 0.0 0.000 50.000 100.000 150.000 200.000 Velocity (cm / sec) Table 3: Theoretical vs. Experimental Projectile Motion The percent difference between the two slopes is given by: 𝑥 − 𝑥 0.4515 − 0.3649 𝑡 𝑒 100% = 100% = 19.18% 𝑥 0.4515 𝑡 The percentage error between the two slopes is roughly equal to the average percentage error between the ten measured distances. The precision of the devices used for measurement cannot be the only cause for such an error. One possible explanation is instead the effect that air friction has on the ball. This would, however, imply that the theoretical results are incorrect. Verifying the accuracy of the experimental results would, therefore, become impossible. Considering the minimal amount of manual measurements performed, and the sophistication of the apparatus used to measure initial velocity (millisecond precision photogate), the intrinsic systematic error of ignoring air friction forces may be the cause for the large discrepancy between experimental and theoretical values. Projectile motion is an example of an object having momentum. Momentum, contrarily to what was believed with the Impetus theory, is not a residual force left in an object. Momentum is the result of acceleration due to an impacting net force, affected by the mass of the object itself. This is what is stated in Newton’ s Second Law: 𝑚 𝑎 = 𝐹 Where 𝑚 is mass, 𝑎 is acceleration, and 𝐹 is net force. Conclusion Throughout the experiment, a mathematical correlation between the initial velocity of the ball and the distance it travelled was found. Although it yielded a relatively high percent error compared to its theoretical counterpart, it was nonetheless sound in form.