Free-Fall Motion. All objects near the surface of the Earth experience a downward acceleration caused by gravity. The

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1 Author: Bob Leonard Lab partner: Rob Horne 10/6/2012 Free-Fall Motion Abstract All objects near the surface of the Earth experience a downward acceleration caused by gravity. The acceleration due to gravity is the most important physical phenomena governing objects near the surface of the Earth. Knowing the magnitude of this acceleration and the effect it has on objects is crucial in the fields of Engineering and Physics. In the first part of the lab we studied the motion of an object dropped from rest. From these results we found that the value of g to be approximately Once the acceleration due to gravity was measured, we studied two-dimensional projectile motion. We qualitatively observed that the motion of an object in free-fall is symmetric about the point where the object reaches its maximum height. We also found that the measured maximum height and range of an object launched from an angle near the surface of the Earth could be adequately predicted by assuming a constant acceleration of g in the vertical direction and no acceleration in the horizontal direction. Procedure and Data For the first portion of this lab, I dropped a coin from a known height above the ground. The initial heights were measured using marks made on the wall by our teacher. We estimate the error in the initial height to be ±0.5 in. My lab partner, Rob, used a stopwatch to measure the amount of time it took for the coin to fall to the ground. To ensure the stop watch started at the appropriate time, I counted down from three before dropping each coin. The drop time was recorded three times for each of the following heights: 88, 77, 66, 55, 44, 33. The results of this experiment are summarized Table 1.

2 Height (in.) ±0.5 in Height (ft.) Time 1 (s) Time 2 (s) Time 3 (s) Average Time (s) Std. Dev. (s) Table 1: Drop time for a coin dropped from rest from various heights. For the second part of the lab a spring loaded Pasco Mini-Launcher was used to launch a steel ball bearing with a reliably constant initial speed. The launcher was clamped to the table in so that the ball left the launcher at the same height as the surface of the table. Rob loaded the launcher by pushing the ball bearing into the launcher until it clicked. He then oriented the launcher so that it launched the ball bearing at an angle of 10 above the horizontal, using a protractor and plumb line to measure the orientation of the launcher. I positioned myself at the approximate place where I expected the ball to land. When Rob and I were both ready, Rob launched the ball bearing using the Mini-Launcer, and I marked the location where the ball bearing first hit the table. Using a meter stick, I recorded the distance the ball traveled before hitting the table. We repeated this process for initial angles of 15, 30, 45, 60, 75 ; each angle measured to within ±1. We also repeated this experiment with larger initial velocities, which were achieved by pushing the ball bearing into the launcher until we heard two clicks. Our results are summarized in table 2.

3 Angle (degrees) ±1 1-click Range (mm) ±5mm 2-click Range (mm) ±5mm Table 2: Range of the metal ball bearing when launched from various initial angles. Calculations The displacement of an object accelerating from rest is given by: Δy = 1 2 a(δt)2 Therefore by plotting the initial height of our coin, Δy, versus the time (Δt) 2, we expect to get a straight line. Moreover, the slope of this line should equal m = 1 a, where a is the acceleration of the coin. Fitting this 2 data to a straight line (see Graph 1), we obtain a slope of m s2, and an acceleration due to gravity of 10.2 m s 2. This acceleration is within 10.2% of the accepted value of 9.80 m s 2.

4 Initial Height (meter) y = x Δt 2 (s 2 ) Graph 1. Δy versus (Δt) 2 for the coin dropped from rest. During the second part of the lab, we studied the motion of a ball bearing launched from a known angle. From the pre-lab assignment, we found that the distance an object will travel before returning to its initial height is given by: range = v 0 2 a sin(2θ) This equation was obtained by assuming a constant acceleration in the vertical direction, and no acceleration in the horizontal direction. The data we acquired for the range of the ball bearing resembles a sin(2θ) shape. Specifically, the maximum range appears to occur near 45. Furthermore, extrapolating our data suggests that the range will go to zero near 0 and 90 suggesting that the range is indeed described by a sin(2θ). Lastly, we can see that increasing the initial velocity of the ball bearing does not change the overall shape of the graph, instead it only scales the data by a nearly constant multiplicative factor as our equation predicts.

5 Range (cm) Initial Angle (degree) Graph 2. The range of the ball bearing versus the initial angle. Error analysis Although the measured acceleration due to gravity from the first part of the lab closely matched the experimental value, there were several important sources of error. One source of error is caused by the limitations in the precision of our measuring devise, this error mostly effected our error in the initial height, since the stopwatch was able to measure times to 0.01 second precision. Even through the stopwatch was capable of measuring time to within 0.01 seconds, another source of error is caused by our inability to start and stop the stopwatch at precisely the right times. This is a source of both random and systematic error. Each time was measured 3 times, the standard deviation provides us with a useful measure of the random error caused by starting and stopping the watch. However, systematic errors, which are not detected by the standard deviation, can also affect our results. For example if the person using the stopwatch consistently starts the watch late, or stops the watch too soon, the resulting systematic error would cause us to overestimate the acceleration due to gravity.

6 The errors in the second part of the lab were likely less significant. While some error is caused by limitations in the precision of our measuring device, a meter stick, the largest source of error comes from estimating the position where the ball bearing lands. Another source of disagreement between the theory and our measured data is the omission of drag from the model we used to predict the motion of the ball bearing. Conclusion By plotting height versus time squared for an object dropped from rest an observing that the resulting graph is a straight line, we can conclude that objects near the surface of the Earth fall to the ground with a constant acceleration. Moreover, using the slope of this graph, we found that the acceleration due to gravity is approximately 10.2 m s 2. This value is in close agreement with the theoretically accepted value of 9.80 m s 2. In the second part of the lab, by plotting the range versus the initial angle of the launcher, we were able to qualitatively verify that the range is accurately described the theoretically predicted equation, v 0 2 sin(2θ). a