In this investigation, a basketball was dropped from a certain height (2.24m), and its displacement was measured over a short period of time (approximately 5 seconds) using an ultrasound distance sensor which was fixed directly above. The basketball weighed 602g and had a diameter of 24.2 cm. The sampled data was recorded and used to plot various graphs.
Table of Data (1,2):
The recorded data is shown in the table, along with the displacement from the ground, velocity, acceleration, gravitational potential energy, kinetic energy and total energy. Although 99 samples were taken altogether, the last 8 have been discarded as the readings after that point became erratic.
Displacement/Time graph (3):
This graph shows the displacement of the basketball from the sensor. Each separate bounce is clear to see.
Displacement/Time graph (4):
This graph also shows displacement, but it shows the displacement from the ground, which is easier to interpret. It is noticeable that at the point where the ball should be touching the ground, it is not quite. This is because, although the ball bounced mainly vertically, the floor was not perfectly flat and so an element of sideways movement was introduced. Therefore the ball moved slightly away from the sensor horizontally.
Velocity/Time graph (5):
This graph shows the velocity of the basketball. The values for its velocity were worked out by calculating the gradient of the displacement/time graph at every point. As is shown by the straight lines, the acceleration is constant during the time when the ball is in the air. This also shows that the air resistance was constant throughout the bounce, or the changes were so negligible that they made no visible difference
Acceleration/Time graph (6):
This graph shows the acceleration of the basketball. The values for acceleration were worked out by calculating the gradient of the velocity/time graph at every point. The acceleration is constant, at about -9.6m/s, except for when the ball is touching the floor, when the acceleration is radically different.
GPE/Time graph (7):
This graph shows the ball’s gravitational potential energy. This is worked out using the equation GPE = mass x g x height. As there is a direct relationship between GPE and height (g and mass being constants), the graph has exactly the same shape as the displacement graph.
KE/Time graph (8):
This graph shows the kinetic energy of the ball. It is worked out using the equation KE = 1/2 x mass x velocity2. Some of the peaks on this graph (where the ball hits the ground) are ‘split in two’. If there was a higher sampling rate, all of the peaks would be split like this, and the split would extend all the way to 0. However, during the actual bounce, the ball is only at 0m/s for an immeasurably short time. Therefore, if samples are taken either side of this point when the ball is travelling at its fastest, it can give the appearance that the ball never slows down when it hits the ground.
Total Energy/Time graph (9):
This graph shows the total energy of the ball. It can be worked out by adding the GPE and KE together. The only time when this does not work is during the bounce when the ball is touching the floor, and the energy is converted into elastic strain energy. From the graph it is easy to see how the ball loses some of its energy on each bounce, as is shown by the ‘steps’.
The air resistance can be calculated by resolving all the different forces. The equation gained from resolving can then be solved. (a is taken from a point on the acceleration graph)
R ( ) : mg + F = ma
0.602x-9.8 + F = 0.602x-9.6
-5.8996 + F = -5.7792
F = 0.1204N
Energy Lost On Each Bounce:
This can be worked out simply by finding the change in GPE, which at the peak of the bounce is equal to the total energy, from one bounce to the next.
1st to 2nd: change in GPE = 10.49888 – 7.48286
Percentage of energy lost = 28.727%
2nd to 3rd: change in GPE = 7.48286 – 5.2976
Percentage of energy lost = 29.204%
3rd to 4th: change in GPE = 5.2976 – 3.89494
Percentage of energy lost = 26.477%
It is apparent that the same proportion of energy is lost on each bounce, approximately 28%.