Experiment: Measure the Velocity of Various Objects in Freefall
Purpose:
The purpose of this experiment was to measure the velocity of various objects in
freefall. Upon completion of this lab one will be able to calculate the values for
acceleration and the effect of air resistance.
Apparatus:
2m clear plastic tube
Acculab sonic ranger (speed of sound = 343 m/s period = 0.03 seconds)
SensorNet software
Macintosh computer with system 6.07 or later
Blank 3.5 HD Macintosh formatted disk
Electronic mass scale, calipers
Various objects to drop
Procedure
For the first part of Experiment 1 we dropped three objects of the same shape, but of
varying mass. To promote accuracy, we performed three trials for each object. The first
object we dropped was a 301 gram ball of radius 1.85 cm, followed by a 225 gram ball with
radius 1.82 cm, and finally a 20 gram ball of radius 1.81 cm. From this information we
were able to derive the mass densities of the objects.
mass density = mass/((4/3)pR3 )
We recorded the relevant data and evaluated the average acceleration by means of the
slope of the velocity graph. From the following equation we determined the magnitude of
the net force acting on the object:
Force net = mass x acceleration = Force gravity + Force air resistance
Next, we calculated the errors and uncertainties:
mean = 1/N [t1 + t2 + ... + tN] ? 1/N S ti
s = [1/N S (mean - ti )2]1/2
For the next part of Experiment 1, we simply conducted three trials for a ping pong ball
of mass 20 grams and radius 1.81 cm. We followed the same basic procedure as above.
Upon completion of three trials, we compared the data collected for the light object with
that of a heavy object.
The method used in Experiment 2 is a lot like that used in the previous experiment. Only
now we are concerned with the velocity and acceleration during a jump. It took us
several attempts to finally achieve three data samples in which the ball did not hit the
wall of the tube. We evaluated the acceleration at several different key points and
recorded our data. Then we calculated the errors using the equations stated above.
Relevant Data
On the following page are three representative samples of raw data from one trial for
each of the experiments we conducted. The slope of the velocity graph, which can be
calculated by the computer, is equivalent to the average acceleration. The mass density
was calculated using the equation stated in the Procedure. The mean values of the slope
for the three combined trials is calculated using the mean formula. The standard
deviation is found using the standard deviation formula.
Sample Calculations (using Mass = 301 g; Radius = .0185 m)
Mass Density
mass density = mass/((4/3)pR3 )
mass density = 301 g/((4/3)p(.0185)3 )
mass density =1.13 ? 107
Mean Average Acceleration
mean = 1/N [t1 + t2 + ... + tN] ? 1/N S ti
mean = 1/3 [14.605 + 6.394 + 5.634] ? 8.877
Standard Deviation of the Mean
s = [1/3 S (8.877 - {14.605, 6.394, 5.634})2]1/2
s = 2.344
Average Acceleration
average acceleration = mean average acceleration ? standard deviation
average acceleration = 8.877 ? 2.344 m/s2
Experiment 1.1
Mass = 301 g
Radius = .0185 m
Mass Density = 1.13 ? 107
Mean of the Acceleration = 8.877
s = 2.344
Time (s) Distance (m) Dist (vel) Dist (accel) Vel (m/s)
1.32 0.562 2.922 2.922
1.35 0.663 3.368 14.887 3.368
1.38 0.773 3.647 9.304 3.647
1.41 0.889 3.889 8.064 3.889
1.44 1.014 4.168 9.304 4.168
1.47 1.144 4.317 4.962 4.317
1.5 1.285 4.708 13.026 4.708
1.53 1.433 4.913 6.823 4.913
1.56 1.595 5.415 16.748 5.415
1.59 1.767 5.732 10.545 5.732
1.62 1.95 6.104 12.406 6.104
Mass = 225 g
Radius = .0182 m
Mass Density = 8.91 ? 106
Mean of the Acceleration = 12.617
s = 1.392
Time (s) Distance (m) Dist (vel) Dist (accel) Vel (m/s)
0.69 0.585 2.922 2.922
0.72 0.686 3.35 14.267 3.35
0.75 0.8 3.815 15.507 3.815
0.78 0.909 3.647 -5.583 3.647
0.81 1.033 4.131 16.128 4.131
0.84 1.168 4.503 12.406 4.503
0.87 1.307 4.615 3.722 4.615
0.9 1.458 5.024 13.647 5.024
0.93 1.62 5.415 13.026 5.415
0.96 1.791 5.713 9.925 5.713
0.99 1.97 5.936 7.444 5.936
Mass = 20 g
Radius = .0181 m
Mass Density = 8.05 ? 105
Mean of the Acceleration = 13.672
s = 1.593
Time (s) Distance (m) Dist (vel) Dist (accel) Vel (m/s)
0.57 0.555 2.903 2.903
0.60 0.650 3.182 9.304 3.182
0.63 0.755 3.480 9.925 3.480
0.66 0.868 3.759 9.304 3.759
0.69 0.988 4.020 8.684 4.020
0.72 1.069 2.680 -44.661 2.680
0.75 1.066 -0.093 -92.424 -0.093
0.78 1.100 1.154 41.560 1.154
0.81 1.108 0.242 -30.395 0.242
0.84 1.728 20.675 681.087 20.675
0.87 1.903 5.825 -494.998 5.825
Experiment 1.2
Lead Ball
Mass = 225 g
Radius = .0182 m
Mass Density = 8.91 ? 106
Mean of the Acceleration = 12.617
s = 1.392
Time (s) Distance (m) Dist (vel) Dist (accel) Vel (m/s)
0.69 0.585 2.922 2.922
0.72 0.686 3.35 14.267 3.35
0.75 0.8 3.815 15.507 3.815
0.78 0.909 3.647 -5.583 3.647
0.81 1.033 4.131 16.128 4.131
0.84 1.168 4.503 12.406 4.503
0.87 1.307 4.615 3.722 4.615
0.9 1.458 5.024 13.647 5.024
0.93 1.62 5.415 13.026 5.415
0.96 1.791 5.713 9.925 5.713
0.99 1.97 5.936 7.444 5.936
Ping Pong Ball
Mass = 2 g
Radius = .0180 m
Mass Density = 8.18 ? 106
Mean of the Acceleration = 9.862
s = 2.955
Time (s) Distance (m) Dist (vel) Dist (accel) Vel (m/s)
0.81 0.511 2.456 2.456
0.84 0.600 2.977 17.368 2.977
0.87 0.694 3.145 5.583 3.145
0.90 0.799 3.480 11.165 3.480
0.93 0.908 3.629 4.962 3.629
0.96 1.024 3.889 8.684 3.889
0.99 1.151 4.206 10.545 4.206
1.02 1.280 4.317 3.722 4.317
1.05 1.416 4.522 6.823 4.522
1.08 1.559 4.764 8.064 4.764
1.11 1.709 5.006 8.064 5.006
1.14 1.859 5.006 0.000 5.006
1.17 1.985 4.206 -26.673 4.206
Experiment 2
Tennis Ball on way down first time
Average Acceleration = 9.367 m/s2
Time (s) Distance (m) Dist (vel) Dist (accel) Vel (m/s)
0.84 0.575 2.810 2.810
0.87 0.678 3.424 20.470 3.424
0.90 0.779 3.387 -1.241 3.387
0.93 0.893 3.796 13.647 3.796
0.96 1.014 4.020 7.444 4.020
0.99 1.147 4.429 13.647 4.429
1.02 1.283 4.541 3.722 4.541
1.05 1.426 4.764 7.444 4.764
1.08 1.579 5.099 11.165 5.099
1.11 1.742 5.452 11.786 5.452
1.14 1.911 5.620 5.583 5.620
Tennis Ball just after hitting the floor
Average Acceleration = -0.620 m/s2
Time (s) Distance (m) Dist (vel) Dist (accel) Vel (m/s)
1.20 1.911 -2.401 -2.401
1.23 1.796 -3.833 -47.763 -3.833
1.26 1.694 -3.387 14.887 -3.387
1.29 1.600 -3.145 8.064 -3.145
1.32 1.518 -2.717 14.267 -2.717
1.35 1.444 -2.494 7.444 -2.494
Tennis Ball near the top of its trajectory
Average Acceleration = 14.577 m/s2
Time (s) Distance (m) Dist (vel) Dist (accel) Vel (m/s)
1.56 1.176 -0.577 -0.577
1.59 1.173 -0.093 16.128 -0.093
1.62 1.182 0.298 13.026 0.298
Tennis Ball on way down second time
Average Acceleration = 9.361 m/s2
Time (s) Distance (m) Dist (vel) Dist (accel) Vel (m/s)
1.65 1.197 0.484 0.484
1.68 1.220 0.782 9.925 0.782
1.71 1.253 1.098 10.545 1.098
1.74 1.291 1.265 5.583 1.265
1.77 1.340 1.638 12.406 1.638
1.80 1.398 1.917 9.304 1.917
1.83 1.464 2.196 9.304 2.196
1.86 1.539 2.512 10.545 2.512
1.89 1.623 2.791 9.304 2.791
1.92 1.718 3.164 12.406 3.164
1.95 1.818 3.331 5.583 3.331
1.98 1.925 3.573 8.064 3.573
Analysis
After analyzing our data we discovered some information concerning the relationships
between different characteristics and properties of freefall. The graph of Acceleration
vs. Mass, depicted on the following page, reveals no relationship between mass and
acceleration. After completion of Experiment 1.1, our data showed that objects of
lighter mass fall with a greater acceleration than those of greater mass. However, when
we performed Experiment 1.2, our data for the ping pong ball did not follow the same
trend as our data from Experiment 1.1. Upon careful evaluation, we discovered the
relationship between mass density and acceleration. Objects of great density, such as
the 301 gram ball, fall with less acceleration than objects of little density such as the
ping pong ball.
Experiment 2 revealed many things about acceleration. The acceleration downward is very
close to the value of 'g', approximately 9.8 m/s2. This value of the average downward
acceleration is constant for both the first time down and the second time down. We found
the value of the average acceleration during the first descent to be 9.367 m/s2. This
value is very close to the value we found for the second descent, 9.361 m/s2. The
difference between these values and 9.8 m/s2 can be attributed to air resistance. We
also found the value for the acceleration just after hitting the floor to be -0.620 m/s2.
The value of the average acceleration near the top of the trajectory was found to be
14.577 m/s2.
Discussion and Conclusion
From this experiment, one learns how to measure the velocity of various objects in
freefall. From this data, one can then calculate the values for acceleration and air
resistance. This experiment relates directly to everyday situations. For example when
one dribbles a basketball, the acceleration of the object becomes negative. Acceleration
that at one time was direct towards the earth, is now directed in the opposite direction.
From our calculations we concluded that as mass density increases, the average
acceleration decreases and approaches 'g.' We also concluded that the downward
acceleration is not related to air resistance. However we did find a relationship
between velocity and the force of resistance. As the velocity of the object increases,
the force of resistance decreases. We also found that the mass of an object has no
effect on the downward acceleration of the object in freefall. From this experiment one
discovers the relationships between different properties of an object.
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