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Sophie Wang xw2218 Physics 1493 Experiment 1: Velocity, Acceleration and G Introduction In this experiment, we are hoping to characterize the motion of an cart under uniform acceleration by taking measurements of velocity with no acceleration and with acceleration due to gravity on an inclined plane. With these measurements, we derived an experimental value of g. The value that we experimentally determined for g was 8.3 ± 1.1 m/s2 which differs from the expected value of 9.8 m/s2 by 15.3%. Although our experimentally determined value is not in perfect agreement with the accepted value, our value was within two standard errors and is thus in good agreement with the accepted value. Procedure To take measurements of constant velocity and constant acceleration, we used a sonic ranger to track a cart moving across a frictionless air track. The first part of the experiment was performed with a level air track. Before beginning to take data, we attempted to level the track using sheets of paper. Then, we set the cart in motion with a gentle push; it travelled to the left end of the track where it bounced off an elastic bumper, causing it to travel back towards the right. During this time, the sonic ranger at the right end of the track was sending 20Hz sound waves, using the time it takes for the waves to be reflected back to determine the cart's position and velocity. The data acquisition PC connected to the sonic ranger was then able to generate graphs, plotting position and velocity versus time. Once we performed ten trials under constant velocity (trying to push the cart with a constant force, and starting each trial at 150cm to reduce error), we used the data acquisition PC to perform linear fits on the increasing and decreasing portions of the position graph. The slopes of these two linear fits gave us values for initial and final velocity, respectively. The second part of the experiment consisted of measurements of acceleration under five different angles of inclination. We inclined the air track with metal shims under its right leg, and let the cart travel down the plane, bounce off the elastic bumper, travel back up the plane, stop and the go back down. Using the sonic ranger, we measured the velocity and position of the cart between the first time it hit the elastic bumper and the second. We performed 50 trials, 10 for each metal shim we added below the air track. Again, to reduce error, we started the cart at 150cm for each trial. To determine acceleration under these conditions, we performed a quadratic fit on the position graph (and took the coefficient A of At^2 to be 1/2* acceleration, according to the kinematics equation) and a linear fit on the velocity graph, to determine the slope, ax (the acceleration of the cart). We then perform data and analysis and calculation in Microsoft Excel to determine values for the coefficient of restitution and g, the gravitational acceleration. Results Using Microsoft Excel, I entered all data into a spreadsheet and performed calculations. For the coefficient of restitution e, I divided Vfinal by Vinitial for each of the ten trials from the first part of the experiment. Then I calculated the weighted and unweighted means for the set of ten values, as well as associated errors. Below are the values obtained:

Unweighted Mean of Coefficient of Restitution, ej ± σej Standard Deviation of Coefficient of Restituion Weighted Mean of Coefficient of Resitution, ew ±

0.759 ± 0.006 (no units) 0.02 0.757 ± 0.001 (no units)

σew

To determine the acceleration due to gravity, I performed several calculations based on our 50 measured accelerations, ax, obtained from linear fits of the velocity graphs. First, I determined the unweighted mean and standard error of ax, then plotted them with respect to h, the height dispacement caused by adding the metal shims. I then obtained the experimental value of g using the equation g=L*ax*h or multiplying L (1.0m) by the slope obtained from performing LINEST on the plot of ax vs h. Height (m) 0.00124 0.00248 0.00372 0.00496 0.00620

mean ax ± σax (m/s^2) 0.0151 ± 0.0007 0.0349 ± 0.0063 0.0370 ± 0.0018 0.0487 ± 0.0006 0.0602 ± 0.0006

m ± σm (s^-2) 0.084 ± 0.011

b± σb(m/s^s) 0.007±0.004

g± σb 8.3 ± 1.1 m/s2

MeanAcceleration vs. Height 0.08

0.07

y =0.0839x +0.0079 R2 =0.9525

Acceleration (m/s^2)

0.06

0.05

0.04

0.03

0.02

0.01

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Height (cm)

To determine the effects of friction, I calculated v1 and v2 (using the weighted mean for e, which is more accurate since it takes into account the different errors of each measurement, giving more weight to more precise measurements) and compared the expected length the cart moves back up the track to the experimental length obtained: h (m)

v1 calc (m/s) e

v2 calc (m/s)

L2 calc (m)

L2 exp(m)

0.00124 0.00248 0.00372 0.00496 0.00620

0.156 0.220 0.270 0.312 0.345

0.118 0.167 0.204 0.236 0.264

0.573 0.573 0.589 0.585 0.573

1.34 1.44 1.09 0.98 1.02

0.757 ± 0.001

Surprisingly, the cart seemed to travel further than it is expected to. The experimental distances travelled up the plane are roughly double what I calculated them to be, which doesn't allow me to calculate friction losses. This discrepancy may be due to a systematic error, which I will discuss in the next section.

Discussion/Conclusion The values I obtained for the weighted and unweighted mean for coefficient of restitution and their associated errors are very close to each other. This likely means that the individual measurements we took for velocity had very similar errors, thus when weighting the mean, we don't see any big changes due to more precise values being weighed more heavily.

Coefficient of Resitution, ej

Coefficient of Resitution vs. Initial Velocity 0.8 0.78 0.76 0.74 0.72 0.7 0.68 0.144

0.18

0.211

0.213

0.222

0.232

0.235

0.289

0.302

0.311

Initial Velocity (m/s)

To determine whether the coefficient of restitution is dependent on initial velocity, I plotted the relationship. It looks as if there is no dependency, which agrees with my expectation. The amount of velocity lost in a collision should not depend on the initial velocity but rather the collision and material properties of the two objects colliding. However, this graph is not totally convincing either way - perhaps with a better spread of initial velocities and more data points, we would be better able to see if there is a trend. The value obtained for g was 8.3 ± 1.1 m/s2. As stated in the introduction, this is within 2 σs of the accepted value of 9.8 m/s2, which means the two values are in 'good agreement'. However, the precision of our measurements are severely limited by the experimental equipment. The air track is said to be frictionless, but may in fact have some small degree of friction (more on friction later) leading to a non-constant velocity. The sonic ranger is also not a perfect instrument because it only delivers sonic pulses 20 times a second - with a higher frequency, we would be able to obtain more precise measurements of position and velocity, which would lead to more accurate linear fits and measurements. Additionally, the metal shims we used to create angles of inclination may not be perfectly uniform and of consistent thickness. This type of inconsistency could also have lead to the inaccuracy of the g value. If the metal shims are inconsistent, this would affect the Δh in the ax vs h graph. Since we used the slope of that line to determine g, a lower Δh (i.e. thinner shims than 1.24mm) would have resulted in a steeper line and a higher value of g that is closer to 9.8. There are a lot of potential sources of error in this experimental set-up. A more precise way to measure g may be to use a simpler setup, like a pendulum, which eliminates several of those sources of error. Although there would still be effects of air resistance, the other sources of error would be eliminated. By looking at the results obtained for expected distance travelled up the plane, it is evident that there may be a systemic error in our experiment. The cart travelled much higher than we expected it to. These two anomalies can be explained by a single systematic error. My hypothesis is that our air track was not level to begin with. Before we started taking measurements, we attempted to level the track, but did not do a great job of it because the cart was not stationary on the track. Instead, it moved back and forth on the track. We chalked that down to the momentum we imparted when placing the cart down; however, we never left the cart

on the track long enough to determine if there was indeed a slight tilt. I believe, based on the calculations of l expected and l experimental, that our air track was inclined so that the right end was slightly lower than the left. If the right end is lower, that means that when adding the shims, the height difference was not 1.24mm for example. If the right end was at -0.5mm (taking an arbitrary number for example), then adding a 1.24mm shim would only make the height difference 0.84mm between the left and right ends of the track. Thus the angle produced is also not as big as expected when the track is level, since sin theta = h/L. Therefore, if the angle is actually smaller than the assumed angle used in calculations, then the cart would travel higher than l calculated. This is a plausible explanation because a 0.5 mm deviation, causes the angle to decrease from 0.071 to 0.044, which leads to a change in l calculated from 0.572 m to 0.92 m nearly a two-fold increase in distance travelled from a minute difference in height. Thus, it is highly possible that a minute tilt in the air track could have caused the discrepancies in l calculated and l experimental. Because of that discrepancy, I am unable to calculate the loss due to the friction. However, if hypothetically, the cart travelled less than we had calculated, that would be evidence of friction absorbing some of the cart's energy as it travels along the air track. One way to quantify the effect of friction would be to determine the potential energy of the cart after it stops after travelling up the plane and compare it to the potential energy (or total energy, since it had no kinetic energy at that point) when you first released the cart. So the loss due to friction would be equal to, mg (hi-hf)/ mghi, which would reduce to (hi-hf)/hi,. This is the loss due to friction as a fraction of initial energy in the system. In conclusion, the values for g obtained were in good agreement with the accepted value for acceleration due to gravity, while the distances travelled were in disagreement due to a possibility of an initial tilting of the air track.

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Unweighted Mean of Coefficient of Restitution, ej ± σej Standard Deviation of Coefficient of Restituion Weighted Mean of Coefficient of Resitution, ew ±

0.759 ± 0.006 (no units) 0.02 0.757 ± 0.001 (no units)

σew

To determine the acceleration due to gravity, I performed several calculations based on our 50 measured accelerations, ax, obtained from linear fits of the velocity graphs. First, I determined the unweighted mean and standard error of ax, then plotted them with respect to h, the height dispacement caused by adding the metal shims. I then obtained the experimental value of g using the equation g=L*ax*h or multiplying L (1.0m) by the slope obtained from performing LINEST on the plot of ax vs h. Height (m) 0.00124 0.00248 0.00372 0.00496 0.00620

mean ax ± σax (m/s^2) 0.0151 ± 0.0007 0.0349 ± 0.0063 0.0370 ± 0.0018 0.0487 ± 0.0006 0.0602 ± 0.0006

m ± σm (s^-2) 0.084 ± 0.011

b± σb(m/s^s) 0.007±0.004

g± σb 8.3 ± 1.1 m/s2

MeanAcceleration vs. Height 0.08

0.07

y =0.0839x +0.0079 R2 =0.9525

Acceleration (m/s^2)

0.06

0.05

0.04

0.03

0.02

0.01

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Height (cm)

To determine the effects of friction, I calculated v1 and v2 (using the weighted mean for e, which is more accurate since it takes into account the different errors of each measurement, giving more weight to more precise measurements) and compared the expected length the cart moves back up the track to the experimental length obtained: h (m)

v1 calc (m/s) e

v2 calc (m/s)

L2 calc (m)

L2 exp(m)

0.00124 0.00248 0.00372 0.00496 0.00620

0.156 0.220 0.270 0.312 0.345

0.118 0.167 0.204 0.236 0.264

0.573 0.573 0.589 0.585 0.573

1.34 1.44 1.09 0.98 1.02

0.757 ± 0.001

Surprisingly, the cart seemed to travel further than it is expected to. The experimental distances travelled up the plane are roughly double what I calculated them to be, which doesn't allow me to calculate friction losses. This discrepancy may be due to a systematic error, which I will discuss in the next section.

Discussion/Conclusion The values I obtained for the weighted and unweighted mean for coefficient of restitution and their associated errors are very close to each other. This likely means that the individual measurements we took for velocity had very similar errors, thus when weighting the mean, we don't see any big changes due to more precise values being weighed more heavily.

Coefficient of Resitution, ej

Coefficient of Resitution vs. Initial Velocity 0.8 0.78 0.76 0.74 0.72 0.7 0.68 0.144

0.18

0.211

0.213

0.222

0.232

0.235

0.289

0.302

0.311

Initial Velocity (m/s)

To determine whether the coefficient of restitution is dependent on initial velocity, I plotted the relationship. It looks as if there is no dependency, which agrees with my expectation. The amount of velocity lost in a collision should not depend on the initial velocity but rather the collision and material properties of the two objects colliding. However, this graph is not totally convincing either way - perhaps with a better spread of initial velocities and more data points, we would be better able to see if there is a trend. The value obtained for g was 8.3 ± 1.1 m/s2. As stated in the introduction, this is within 2 σs of the accepted value of 9.8 m/s2, which means the two values are in 'good agreement'. However, the precision of our measurements are severely limited by the experimental equipment. The air track is said to be frictionless, but may in fact have some small degree of friction (more on friction later) leading to a non-constant velocity. The sonic ranger is also not a perfect instrument because it only delivers sonic pulses 20 times a second - with a higher frequency, we would be able to obtain more precise measurements of position and velocity, which would lead to more accurate linear fits and measurements. Additionally, the metal shims we used to create angles of inclination may not be perfectly uniform and of consistent thickness. This type of inconsistency could also have lead to the inaccuracy of the g value. If the metal shims are inconsistent, this would affect the Δh in the ax vs h graph. Since we used the slope of that line to determine g, a lower Δh (i.e. thinner shims than 1.24mm) would have resulted in a steeper line and a higher value of g that is closer to 9.8. There are a lot of potential sources of error in this experimental set-up. A more precise way to measure g may be to use a simpler setup, like a pendulum, which eliminates several of those sources of error. Although there would still be effects of air resistance, the other sources of error would be eliminated. By looking at the results obtained for expected distance travelled up the plane, it is evident that there may be a systemic error in our experiment. The cart travelled much higher than we expected it to. These two anomalies can be explained by a single systematic error. My hypothesis is that our air track was not level to begin with. Before we started taking measurements, we attempted to level the track, but did not do a great job of it because the cart was not stationary on the track. Instead, it moved back and forth on the track. We chalked that down to the momentum we imparted when placing the cart down; however, we never left the cart

on the track long enough to determine if there was indeed a slight tilt. I believe, based on the calculations of l expected and l experimental, that our air track was inclined so that the right end was slightly lower than the left. If the right end is lower, that means that when adding the shims, the height difference was not 1.24mm for example. If the right end was at -0.5mm (taking an arbitrary number for example), then adding a 1.24mm shim would only make the height difference 0.84mm between the left and right ends of the track. Thus the angle produced is also not as big as expected when the track is level, since sin theta = h/L. Therefore, if the angle is actually smaller than the assumed angle used in calculations, then the cart would travel higher than l calculated. This is a plausible explanation because a 0.5 mm deviation, causes the angle to decrease from 0.071 to 0.044, which leads to a change in l calculated from 0.572 m to 0.92 m nearly a two-fold increase in distance travelled from a minute difference in height. Thus, it is highly possible that a minute tilt in the air track could have caused the discrepancies in l calculated and l experimental. Because of that discrepancy, I am unable to calculate the loss due to the friction. However, if hypothetically, the cart travelled less than we had calculated, that would be evidence of friction absorbing some of the cart's energy as it travels along the air track. One way to quantify the effect of friction would be to determine the potential energy of the cart after it stops after travelling up the plane and compare it to the potential energy (or total energy, since it had no kinetic energy at that point) when you first released the cart. So the loss due to friction would be equal to, mg (hi-hf)/ mghi, which would reduce to (hi-hf)/hi,. This is the loss due to friction as a fraction of initial energy in the system. In conclusion, the values for g obtained were in good agreement with the accepted value for acceleration due to gravity, while the distances travelled were in disagreement due to a possibility of an initial tilting of the air track.

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