Hypothesis
I think that as temperature increases, the average force acting on the ball by the ground increases. Since when temperature increases, the rebound height of the ball will be higher. I argue that if the ball rebound higher, the net force acted on the ball by the ground must also be greater.
According to the formula the impulse is
As ,
If is greater with impact time being constant, the average must be greater. When temperature of the squash ball is low, it is quite soft and easy to be deformed. The impact time is hypothesized to be longer. As temperature increases, the squash ball will become more rigid and deform less. The impact time is hypothesized to be shorter. Since, when the impact time is smaller with net force being constant, the average force must be greater. Therefore, with the two effects, the average force of the impact is hypothesized to be greater when temperature increases.
Method and materials.
Experiment (a)
Since there is originally no any equipment that can measure the extremely short impact time, therefore I had to develop several ideas to measure the time. These are the six possible solutions:
- Stroboscopic photos
The negative is put under long exposure. And the experiment is supposed to be performed in a dark room otherwise the negative will be over expose. Stroboscope is needed to give the flashes at a very high frequency. Under which, the images of the falling ball including the impacting period will be taken. We will then count the numbers of images that are touching the ground (impact period). With the frequency shown on the stroboscope, we can then calculate the impact time of the squash ball.
However there is limitation of the experiment. The impacting images may pack too close to each other that we cannot distinguish the number of them and fail to calculate the correct value.
Moreover stroboscopic photos are difficult to be taken well. It requires skill to control the exposure so that the photos taken will not be too bright or too dark for observation. Therefore the suggestion was abandoned.
- Ultrasonic position sensor (UPS)
Place a UPS on the ground; drop the ball from certain height to it. The UPS is connected to a computer for receiving data. A graph of distance against time will be plotted automatically. By observing the length of time when the distance is at zero, we can know the impact time of the ball.
However, later I acknowledged that the speed of the ultrasonic waves is not fast enough to measure the fast dropping object to give accurate results. Therefore the suggestion was abandoned.
- Conduction sensor
Fix a piece of foil on a dense plate (cutting board), on the surface place another piece of foil closely but without touching the first one. Both foils are connected to a scalar timer with wires. The ball is then dropped onto the upper foil, pressing the foil and closing the circuit. When the ball rebounds, the upper foil releases and disconnects the circuit. The impact time can be indirectly collected from the conduction time. As this experiment was easier to perform, I used the set up to find a rough impact time of about 0.01s~0.05s. This result can be then used as a assumption value for other suggestion.
However, it was suggested that the upper foil may obstruct the falling speed of the ball. This leads to an experimental error of the results. Moreover after the ball rebounds and leaves the upper foil, the foil may still in touch with the lower foil due to deformation. The impact time we get may be over estimated.
- Light sensor
Set up a light sensor on the table with the light beam just situate above the table surface. Then drop the ball to cut the beam. The time that the light sensor obtains is the impact time.
However, as the light beam has finite thickness, it is not accurate enough to measure the impact time. The ball may cut the beam too early and leave too late which over estimate the impact time. Furthermore, it is difficult to ensure the ball drop exactly to the light beam by its lowest point. The results may not be accurate.
- Formula and calculation
First we need to measure the dropping height, e.g. A cm. Then we drop the ball and at the same time start to count the time using a timer. When the ball rebounds to the highest point we stop the timer and at the same time record the highest point it reaches, e.g. B cm. let the total time for the process be C seconds. From the formula s=ut +(1/2) at2. We then substitute distances B and C to find out the time need for dropping and rebounding.
But deficiencies are still being found for this alternative. There is reaction time error in working the timer. The reaction error is even larger than the impact time. Also, the highest point the ball reaches may not be accurately detected. So the measurement is considered not working.
- Digital-video camera approach
Use the camera to take the impacting images of the ball. Then replay the film to find out the time of impact. As we found out that the impact time is around 0.03
Final decision
After series of consideration, I made the final choice to use the option 3. Despite its limitation that may lead to over estimation of results, I found the problems that may occur in No.3 least essential. Moreover stretching the upper foil a little can reduce the deformation of the upper foil. So this measuring method was selected.
Experimental set up.
In the experiment I prepared the following material for the setting up.
Materials
Kettle, clamp, chopsticks, squash balls, stand, towel, scalar timer, aluminium foil, aluminium tape, plastic tape, wire, clips, cutting board, a pack of unused paper card.
Methods
First of all, Impact Time Measuring Device (ITMD) was made as core of the set up:
Aluminium foil was stuck to the cutting board until its upper surface was completely filled up by the tape. I then check the conduction of the foil to ensure no gaps between each strip of tape. Then an 8x8cm2 hole was made from 10x10cm2 paper card. A piece of 8x9cm2 foil was then stretched on the middle of the hole. Then I used tape to fix the foil on two ends of the hole. The paper card with the foil on top was put onto the upper surface (with foil) of the cutting without the two piece of aluminium touching each other. Then both foils were connected to two separated wires with crocodile clips and the wires were connected to the scalar timer. The ITMD was finished.
In order to test if the ITMD was reliable, I performed several dropping test for checking. Firstly I dropped the ball at room temperature of height 140 cm; unfortunately the results each time collected were not consistent. They had differences of about 50% to 200%. Therefore I changed the setting of the paper card. I used a larger piece of foil (9x9 cm2) and stretched it to the four end of the paper hole. The later tests showed improvement as the differences drop to about 20% to 60%. And I thought that it may due the deformation of foil that the two foils still pressed to each other when the ball left. So I stuck another piece of paper card with hole just right beneath the original one. It was done to increase the distance between the two foils by about 0.5mm so that they are more likely to separate after the ball has left. The tests followed were more coherent as their differences were just about 10 % to 20%. Then I varies the dropping height to see if the measurer could detect the time different (room temperature). It showed an increasing trend of impact time when the dropping height increase. That proved that it senses changes.
The kettle was then used to boil the water for heating up squash balls; however it is not convenient to do in this way, so I changed to use a water bath instead. With the water bath, I could then adjust the temperature I want easily. Clamps were used to release the balls instead of the chopsticks. Firstly stand with clamp were put on the lab table. I measured 130 cm from the bottom of the ball in the clamp vertically to the centre of the cutting board. Then the squash balls were first immersed into water of 20.2oC for 10 minutes to ensure the balls were have same temperature as water. Then I used the clamp to take one from the water bath, quickly dried it with towel and transferred to the clamp on the stand, released it to the centre of the paper. Then I repeated the procedure by another nine times to collect ten data at that temperature. In between, I recorded the impact time from the scalar timer. After that I continued the experiment with an increase of 10 oC until it reached 100 oC. For the handling of hot balls, working gloves were needed.
Experiment (b)
Materials
Clamp, squash balls, stand, towel, scalar timer, aluminium foil, aluminium tape, plastic tape, wire, clips, cutting board, a pack of unused paper card, water bath, working gloves.
- Same platform (cutting board with paper card) in experiment (b) was used to make the condition of two experiments more constant. Pieces of blank papers were first placed along the drop ping track of the ball on the side of the lab table. The balls were taken from the water bath of the temperatures as Experiment (b), dried, transferred to clamp and dropped to the cutting board quickly. The highest points it reached after the rebound were marked onto the papers. I repeated ten times for each temperature. Finally measuring tape was used to measure the rebound height of each temperature.
Data Collection
Dropping Height=130.0 + 0.1cm
-
Temperature=20.2 + 0.4 o C
Uncertainty of height=0.1cm Uncertainty of time=0.001s
Mean height = (18+18.2x2+18.4x3+18.6+18.8+19x2)/10 = 18.5 cm + 0.2cm
Uncertainty =
Mean impact time =
Uncertainty =
-
Temperature=30.0+0.4oC
Uncertainty of height=0.1cm Uncertainty of time=0.001s
Mean height = (27.2+27.4x3+27.8x2+28.4x2+28.6x2)/10 = 27.9 + 0.2 cm
Uncertainty =
Mean impact time = (20.9 + 1.3) 10-3s
Uncertainty =
-
Temperature = 40.0+ 0.4oC
Uncertainty of height=0.1cm Uncertainty of time=0.001s
Mean height = (35.8x2+36.3+36.4x5+36.6+36.8)/10= 36.3+0.2cm
Uncertainty =
Mean impact time =( 21+1.1) 10-3s
Uncertainty =
-
Temperature = 50.0+0.4oC
Uncertainty of height=0.1cm Uncertainty of time=0.001s
Mean height = (41.0+44.6+44.8x4+45.0x2+45.4+45.6)/10 = 44.6+ 0.7cm
Uncertainty =
Mean impact time = (20.8+0.9) 10-3s
Uncertainty =
-
Temperature = 60.0+0.4oC
Uncertainty of height=0.1cm Uncertainty of time=0.001s
Mean height = (51.6+51.8x2+52x2+52.2x2+52.6x2+54)/10 = 52.3+0.4cm
Uncertainty =
Mean impact time = average time = (20.4+0.9) 10-3s
Uncertainty =
-
Temperature = 70.0+0.4 oC
Uncertainty of height=0.1cm Uncertainty of time=0.001s
Mean height = (60.4+60.6x2+61.6+61.8+62.6x2+62.8+63.2)/10=61.8+0.4cm
Uncertainty =
Mean impact time= average time = (20.2+1.3) 10-3s
Uncertainty =
-
Temperature = 80.0+0.4 oC
Uncertainty of height=0.1cm Uncertainty of time=0.001s
Mean height = (68.0+68.8+69.2+69.8+70.6+70.8+71.6+72.6+73.6)/10 = 70.4+0.9cm
Uncertainty =
Mean impact time = average time =( 20.5+0.8) 10-3s
Uncertainty =
-
Temperature = 90.0+0.4 oC
Uncertainty of height=0.1cm Uncertainty of time=0.001s
Mean height = (74.6+74.8+76.8+77x2+77.6+77.8+78.4+78.8)/10 = 77.1+0.7cm
Uncertainty =
Mean impact time = average time = (20.7+1.4) 10-3s
Uncertainty =
-
Temperature = 100.0+0.4 o C
Uncertainty of height=0.1cm Uncertainty of time=0.001s
Mean height = (80.0+80.8x2+81.4+82+82.6+82.8x2+83.4+84)/10 = 82.1+0.6cm
Uncertainty =
Mean impact time = average time =(21+0.9) 10-3s
Uncertainty =
Observation
The size of the squash ball increased as temperature increased.
At high temperatures >80 oC, the surface of the squash ball became rough as some of the rubber skin of the squash ball was boil away.
Analysis
The impact time against the temperature:
Unlike my hypothesis, the result of the impact time of the ball showed no obvious change when temperature increased. Moreover the pattern of the trend was not the way I thought where it was hypothesized to increase as temperature increase.
The graph of rebound height against temperature:
The rebound height showed obvious increase as temperature. The results fit with the hypothesis. The rate of increase of rebound height was quite constant from temperature 20 o C to 80 o C. Then it started to decrease from temperature 80 o C to 100 o C.
At the time of impact, the force diagram of the ball is like this:
There were two forces acting on the ball, one is the normal force acted by the ground, the other one is the gravitational force acted by the earth. Since net force of the ball, therefore the force acted by the ground
From the graph of force acted on the ball against temperature. I found that the highest increase rate of force occurring at temperature at around 20-30o C, high increase rate continued from 20-65 o C. The rate of increase slowly decreased as it approached temperature greater than 70 o C and finally showed no change around 100 o C. By drawing a line of symmetry from the top of the trend line, we could observe that the highest average force the ground acting on the ball to be about 10.2N.
Discussion
Average force
As stated , the graph showed a maximum average of ~10.2N. It suggested that if the velocity of the squash ball is hold constant, the average force that can be exerted to a squash ball by a stationary impacting surface will be at maximum when temperature around 95o C.
However the average force will not drop to zero when temperature drops to absolute zero. Since the average force acted on the ball by the ground , it is always >0 because the velocity of the ball is changing. According to the Newton’s first law of motion, every body continues on it state of rest of uniform speed in a straight line unless acted on by a nonzero force. At the point of impact, the ball accelerates upward. The only force that points upward is the normal force by the ground. Although there may be a possibility that the ball drops and sticks to the ground at extreme low temperature, causing the △t to be infinite, but still there is the opposing force acts by the ground again the weight of the ball. So the average force by the ground must at least equal to 9.81x0.023 which is ~0.23N.
In addition, the rate of increase of average force acted by the ground is believed to fall at the lower temperature. As shown in the graph above, the trend line is pointing toward zero at temperature>-273. We have explained that the trend will not be zero even when the temperature is -273. One possible way will be a turning point located somewhere between -273 to 20o C. And if there is a turning at that certain point, the rate of increase must be lower at that point.
Rebound height
The rate of increase of rebound is quite constant until at 80 o C it decreased. Theoretically the trend line will approach 130cm when temperature goes to infinity. However it is not possible because squash ball will melt at high temperature. For the low extreme, the squash may not rebound properly as the low temperature may constrict the plastic layer of squash ball, making it deforms, losing it quality. It is just a prediction and is difficult to perform in the school lab.
Impact time
The impact time showed no relation with the range of temperature set for experiment. The set up may not be sensitive to sense the different.
Relation of force and energy at the impact
The potential energy of the ball was changed to kinetic energy before the impact; some of the energy was lost to the air friction. At the impact, some of the kinetic energy was transferred to heat energy of the ground and the ball. Some of it was transferred to the sound energy. It lost his energy and rebounded to a lower height. Those energy did lose in the impact was transferred to build up the shear modulus (elastic energy) of the ball. The greater the elastic energy is the higher the ball rebound.
Evaluation of the experiment
The experiment was considered successful as the data showed a direct relationship between average force and temperature of the squash ball and a decreasing rate of increase of average force when the temperature increases. However the impact times collected were about the same which contradicted to my hypothesis. They were not even in sequence. That may due to the deficiency of the experimental set up. The foils might still connect together a short time after the ball had left. Although the time is short compared to the impact time, it changed for every impact. Therefore the impact time I got was not in a trend but about the length. it may also due to the insensitivity of the set up. After all I still managed to get the approximate impact time for the calculation of average force.
For the rebound height experiment, it was quite good. There was little problem such as the imprecise way of recording rebounding height by using eye observation.
On the whole, there were many systematic errors in both experiments that may affect the results. For examples, the size of the ball increased as temperature increased, it might have increased the impact time of the ball due to larger impacting area. It was possible to be the reasons for the unsuccessful for the impact time results. This might also affect the rebound height of the experiment.
The foil on the cutting board reduced the velocity before impact. It might have reduced the rebound height and the impact time of the ball. The average force might have been over estimated or under estimated depends on the extent of reduction of the rebound height and impact time.
Heat lost rate increased as temperature of ball increased. That suggested that the rebound height should be at the lower temperature. The rebound heights were over estimated for the higher temperature. The impact times were also affected in a certain degree.
Improvement
Renew the upper foil whenever it deforms to avoid over estimation of impact time. However it may be inconvenient.
Change to another method in measuring the impact time. e.g. light sensor.
For the measuring of the rebound height, we can ask a partner to observe the rebound ball at the same level to improve accuracy. We can also do more repetitions for more data.
Drop the squash balls directly without transferring them to the clamp on the stand. However high delicacy is need to ensure the dropping height is right and not initial force is applied to the ball.
Conclusion
The results of the experiment stated that there are changes of average force acted on the ball by the impact surface with the velocity of ball hold constant. The maximum average force will be reached at temperature around 95oC. This proved that the hypothesis to be true. However the hypothesis for the impact time was not proven to be true as the set up was appropriate enough to measure the data accurately.
Nevertheless, the result still showed the rate of increase in average force of impact at different. By using the data we can know that at what temperature does the squash ball work most effectively with the smallest force given. The data can also be useful for the manufacture of squash ball.
Appendix
