How to calculate energy loss from falling balls? Good evening,
I am doing a science investigation about "Which surface will affect the energy loss of a bouncing basketball the most?" and I am having trouble calculating the energy loss for my data. I will include the data in my investigation. I start by putting a 1 meter measuring stick , putting the basketball at the 1 meter line, having a friend record everything and then dropping the ball and letting it bounce once. I repeat this 5 times for each surface before moving to a different one. After I have gotten all of the videos, I watch them and check what is the data inside of the videos (how far they bounced) and write it down. Now I need to calculate the energy loss.Thank you
 A: Mechanism of energy loss
The principal energy loss when a basketball ball bounces off the ground is most likely due to the non-adiabatic compression of the ball amterial and the air inside the ball - some of the energy is converted to heat and cannot be recovered.
There may be also similar energy loss in the material of the floor/ground, particularly, if we are dealing with an inelastic material, such as dirt.
Experiment design
One could try dropping the ball from the same hight and measuring its maximum elevation after it rebounds from the ground. Presumably, this is what the data in the OP present. There are a few caveats that may skey the results of the experiment, particularly those related to the initial state of the ball, such as:

*

*the pressure inside the ball (which may change after a few trials, be unequal, if different balls are used, or change with temperature and atmospheric pressure, if the expeirments are done on different days)

*the temperature of the ball (a ball that has bounced a few times is likely to be warmer and have different properties).

Data analysis
The data obviously exhibit variation due to the ucnontrollable random factors. We are delaing here with several groups of measurements, where the bounce height can be characterized by its mean and variance. The first statistical methods to use in such a setting are:

*

*ANOVA (ANalysis-Of-VAriance) to establish whether there is difference at all between different types of ground (most probably, tehre is, as the results for Dirt are clear outliers. One may however have difficulty obtaining a significant result, if the Dirt is excluded).

*Pairwise t-tests for more detailed study.

The testing is needed in order to assure that there is actually a significant difference between measurements carried with different types of surface. A rule of thumb (to use instead of testing) is that the difference between the means should be greater than the sum of the standard deviations - otherwise one cannot really claim that two types of ground really rpoduce different rebound. Better results are obtained with more measurements, as thsi reduces the standard deviation.
Energy loss
Assuming that the initial speed of the ball is zero, as in my proposed experimentd esign, the energy loss is
$$\Delta E=mg(h_i-h_f),$$
where $h_i$ is the height from which the ball drops, while $h_f$ is the average hight to which it rebounds.
A: When the ball is released it has no kinetic energy.  When it bounces back up to its highest point it also has no kinetic energy.
So the energy loss is the difference between the gravitational potential energies.
For example for Linoleum the initial gravitational potential energy was $mgh = m\times 9.8\times 1$
where $h$ is the 1m height, $m$ is the mass of the basketball and $g$ is the acceleration due to gravity.
So its about $6.076J$ for a $0.62$kg basketball.
After bouncing the $mgh$ formula is $0.62 \times 9.8 \times 0.79 = 4.8J$
The energy loss for the ball bouncing on Linoleum is $6.076 - 4.8 = 1.276J$.
That's probably what you are meant to do in this experiment, although the energy isn't actually 'lost' it's converted into other forms such as heat.
All the best with it.
A: From the described experiment you don't "only measure the the energy loss of a falling ball". So as already said you must be more specific what you want to measure. The pure energy loss (dissipation) from a falling ball would be only the friction with ambient air during the fall. If you ask "Which surface will affect the energy loss of a bouncing basketball the most?" again you have to define "energy loss" more specific. Let's go throw step by step

*

*the ball is falling, which forces are acting?
gravitational force, friction with ambient air, boyant force of the ball in air if the ball temperature is different to ambient temperture, potentially vibration and rotatation energy (only for a longer fall)


*the ball is in contact with the surface:
you have to model the surface and body and the underlying body(ies) on which the "surface body" is lying. What I mean is e.g. usually a laminate is layed down on a concrete floor inbetween somefleece etc, a ceramic tile is usually fixed with tile cement on the (again) underlying concrete floor etc for the other materials.
Now back to the problem, you have here a classical collision problem of bodies (2 or more realistically 3 ball - surface body - concrete floor - (it goes further here, is the concrete floor on natural groud or is it in a building e.g. on the second floor)).
These can be split into elastic (no energy loss) and inelastic collision (with energy loss). In addition the ball will compress (and the whole underlying syteem) during contact and collision  - air compression and energy losses (vibration, compression) within the ball body skin.
So the "collision system" affects the energy loss overall the most, e.g. a relative sandwich structur of the laminate system vs e.g. ceramic tile system (inelastic vs quite elastic). The surface itself is not so important, it is the material and the mass of which the "surface body" is made of and how the underlying system is made of.


*ball bouncing back:
which forces are acting now ? See ad 1. with focus on the contributing part of the now  oscillating (vibration) ball.

The "overall energy" loss John and Roger answered already, it's just the difference of the potential energies before and afterwards. E means energy, m mass, g gravitational acceleration. Your data astonishing me, I wouldn't expect that a laminate gives you a more elastic collision than a concrete brick or ceramic tile. So analyze how the underground looked like in your experiment to make it comparable.
