s = 1/2at2
where s= displacement, a = acceleration, t = time
From this, displacement can be calculated and does not have to be measured. This can increase our accuracy because using a stopwatch has a smaller percentage error than using a meter ruler. Using computer to calculate can also eliminate our random error. So using a stopwatch is preferred.
The initial falling height of the tennis ball might seem to be important. But from the formula, it is very clear that the dropping height is not important. As long as we start measuring when the tennis ball first hit the ground, our results will not be affected.
Stopwatch seems to be the most suitable apparatus for measuring time. We can start and stop measuring time when we see the ball hitting the ground. Although using our hands will generate a high percentage error, it is still preferred. Time gate is another time measuring equipment in the laboratory. But this is not appropriate. It is due to the fact that time gates are too precise in time. It cannot measure a long period of time nor can it measure a long length.
Data Collection:
Uncertainties of time: ±0.2s
Data Analysis:
Due to the fact that the time is the time it takes to complete the whole displacement, which is double of the height. So we divide time by two to get the time it takes to complete the displacement from the highest point to the ground.
From the graph, we can see there is a relationship between the different rebounces. This is due to the fact that as the ball first hit the gorund, much energy is lost to overcome the significant air resistance and some kinetic energy is changed into heat energy and transferred into the ground. So the rebound heights are not as high as the first one, and they have a decreasing trend.
The rate of decreasing rebound height slows down as the number of rebounces increases. This is because at a smaller height, air resistance is not as significant as before. So less kinetic energy is lost to overcome air resistance. Also, the ball hits the ground with a smaller velocity thus transfer less energy as heat to the ground.
From our starting formula, we have:
s = 1/2at2
where s= displacement, a = acceleration, t = time
which follows the general formula of y = kx
We can see from the formula there should be a relationship of s α t2
We can now plot a graph of s against t2.
We can also see that the graph appears to be a curve. In order to prove this is a graph with the correct formula, we can deduce the formula by:
s = 1/2at2
log s = log(1/2at2)
log s = log(1/2) + log(at2)
log s = log(1/2) + log(a) + log(t2)
log s = log(1/2) + log(9.8) + 2log t
log s = 2log t + log(4.9)
As from above, if we plot log s against log t, we should be able to get a straight line graph with slope 2 and a y-intercept of log 4.9 = 1.7
From the graph, there is a straight trendline. So it has been proved that the results are valid to prove that displacement increases proportionally to (time)2 .
Conclusion:
Displacement increases proportionally to (time)2 .
