Bouncing ball - How many times does it bounce

[Ritzycat]

Problem statement:
I have a 0.0506kg bouncy ball that will I drop at 1.75m. I must predict how many times it will bounce before it comes to a rest.

My work:
We did a lab to predict the percent of energy "lost" from each bounce. After doing some calculations with my data, I found that, on average, 37.9% of the total energy of the ball was "lost" after each bounce. (I verified this number with my teacher and classmates) Note that when I say "lost", I mean converted to thermal energy.

So, I took a similar approach to this problem.

PE_g = mgh = (0.0506kg)(9.8m/s²)(1.75m) = 0.868 J

I created a table.

Number of Bounces - Total Energy of Ball (J)
0 - 0.868
1 - 0.539
2 - 0.335
3 - 0.208
4 - 0.129
5 - 0.0801
6 - 0.0498
7 - 0.0309
8 - 0.0192
9 - 0.0119

We are using the same ball for this part. I maintained my assumption that after each bounce, 37.9% of the total energy of the ball will be converted into thermal energy. However, I must predict how many bounces the ball will bounce before it stops. But since I'm taking a percentage of the previous value will theoretically always have kinetic and potential energy, ie. not all will be converted to thermal. So theoretically it would bounce forever. At what height will the bounces become so small that's its unnoticeable?

For example, after the ninth bounce, the ball will reach a height equal to:

(0.0119 J)/((9.8m/s²)(0.0506kg)) = h = 0.024m... or 2.4cm.

Not sure how far I want to take it before the ball supposedly stops bouncing. I will be graded on how accurately I can predict the # of bounces.

 

[ehild]

You measure the height and energy with some accuracy, with three significant digits in your case. The uncertainty of the energy is 0.001 J. If the calculated energy is less than that, you can consider the bounce stopped.
Instead of calculating the energy after each bounce, you should notice that the energies make a geometric sequence. It is easy to determine when the energy is 0.001 J.

 

[Ritzycat]

Yes, I did notice they made a geometric sequence! I wasn't sure if that was relevant or not at the time, though. I also did not know that the uncertainty is where I should "taper" it off. I'll make a geometric sequence.

 

[rcgldr]

In an idealized situation with a fixed coefficient of restitution between 0 and 1, a ball bounces an infinite number of times in a finite amount of time and travels a finite abount of total distance, with the frequency of bounces approaching infinity as the time approaches the limit of time based on the initial conditions (initial height, and coefficient of restitution). For a real ball, eventually the center of mass of the ball moves so little that the ball ceases to leave the surface that it was once bouncing on, just compressing and expanding vertically for a few more cycles until it stops.

 

[HalfLostAndSearching]

Based on the data and calculations provided, it is difficult to accurately predict the exact number of bounces the ball will make before coming to a rest. This is due to the fact that the theoretical calculations assume perfect conditions and do not take into account external factors such as air resistance and imperfections in the ball's surface.

However, we can estimate that the ball will continue to bounce until the height of each bounce becomes so small that it is practically unnoticeable. This can be seen in the example provided where the height of the ninth bounce is only 2.4cm.

To improve the accuracy of our prediction, we can conduct experiments with the actual bouncy ball and observe its behavior under different conditions. This will allow us to make more informed predictions and take into account external factors that may affect the ball's bouncing behavior.

In conclusion, while we can make theoretical calculations and estimations, the exact number of bounces the ball will make before coming to a rest cannot be accurately predicted without considering all external factors.