From the data provided in this experiment on Interactive Physics, it is reasonable to conclude that as the mass of a counterweight on a trebuchet increases, then the distance its projectile travels should also increase with it. The data proves the hypothesis correct except for the fact that the distance did not increase in a linear path, it was more quadratic. The data followed a general and consistent pattern of increasing in distance when the mass of the counterweight was increased in intervals of 5 Kilograms starting with 20 KG and ending with 200 KG. There were some glitches in this pattern though which may have been due to computer error. The graph of the projectile distances was generally equal to the equation -8.1551E-4x^2 + .304388x + 8.12756 (where x equals the mass of the counterweight) and they were released at a 45 degree angle each time. The graph was believed to be more quadratic than linear because the longer the projectile is in the air the more time gravity has to act against it and push it down to the ground. So the projectile distances would not increase as rapidly once more weight is added and the projectile is in the air longer. These results support previous, historical experiments as well as Newton’s Third Law of Motion. However, if the experiment were to have been done in real life and an actual trebuchet had been built, then there would have been a point where too much weight could be put on and the projectile distance would no longer increase with the mass of the counterweight. This is because of the stress put on the trebuchet by the counterweight causing possible breakage. The procedure in this experiment was fairly sound, though, because the experiment was run in a computer simulation where air friction was able to be neutralized and human error in building was minimized. It was also possible to manipulate the program so that the projectile would stop at the exact moment it hit the ground, and the computer would give the precise distance the projectile traveled. This eliminated any uncertainty in the distances. If the experiment were to have been done in real life and the trebuchet built by hand, then the results would be more prone to human error and there would be more uncertainty. Of course this does not mean that there were no weaknesses in the procedure. For example, the counterweight did not have a full range of motion and would often hit the base of the trebuchet. This was really only evident when the mass of the counterweight was fairly large. If the experiment was to be done over, the fulcrum length would be made smaller. This may be why the distance the projectile travels does not increase with each interval of counterweight masses.

From the data acquired in the second experiment, it is reasonable to conclude that a 45 degree angle of launch is the angle that gives the most distance to the projectile. This data also proved the hypothesis correct. The reason that there were no angles under 45 degrees used is because theoretically, in a program where no air resistance is registered, a 50 degree angle should produce the same results as a 40 degree angle and 55 degree angle should be the same as a 35 (consider the equation distance traveled = (Velocity * time) / cos (angle of launch)s=vt/cosθ). Like in the experiment above, the graphs of all the angles were quadratic in nature. The equation of the 50 degree angle graph was -6.4276E-4x^2 + .23849x + 7.03039, the equation of the 55 degree angle graph was -5.33146E-4x^2 + .1970456x + 4.730941, the equation of the 60 degree graph was -4.0621E-4x^2 +.14637x + 3.401566, and the equation of the 65 degree graph was -3.2198E-4x^2 + .1076695x + 1.20833. This is further proof that a 45 degree angle should yield the biggest results. Since this experiment was basically the same as the previous, the errors and improvements were basically the same. One error, however, that was unique to this experiment was that when the angle got bigger and the mass of the counterweight was not that great, the projectile often did not clear the base of the trebuchet itself, so the distance had to be estimated. This could have caused the results to be skewed. But since this only happened on four separate occasions, and the base was very close to the ground, it was probably not enough to mess up the data. Yet if the experiment were to be done over again, the base would have been built smaller so the projectile distances of these particular examples could have been viewed. Overall, these experiments gave very accurate data due to the fact it was conducted on a computer program. There were some glitches in the data, but the same thing would have happened if the experiment were to be run in real life. So the way the experiment was conducted was probably in the most accurate way possible, and this is the way the experiment would be done if it were to be conducted again.

Perhaps the most interesting thing we noted from analyzing the graphs were that they all seemed to approach a limit as the mass of the counterweight grew infinitely large. When we take the formulas found above and derive them for each angle; by solving the derived equation for zero, we end up with the limit. To begin with the equation for the 50 degree angle, we take the derivative of -6.4276E-4x^2 + .23849x + 7.03039, which is -.00128552x+.23849. When we solve for zero, we see that the limit of the 50 degree angle approaches 185.5 meters. Calculating the rest of the angles we progress downward with approximate limits of:

45 degree angle	186.6 meters
50 degree angle	185.5 meters
55 degree angle	184.8 meters
60 degree angle	180.2 meters
65 degree angle	167.2 meters

This data analysis further confirms our hypothesis that the 45 degree angle has the longest distance if the maximum distance the trebuchet will launch the projectile is greatest at 45 degrees. Although this wasn’t part of our original intent of the experiment, it was an interesting trend that we noticed with the graphs.