Calculating a velocity formula using air resistance

[NickMitch]

Homework Statement

I am conducting an in-class practical investigation where I am investigating whether altering the drop height of a ball will affect the coefficient of restitution.
I have completed all the trials however, the data suggest that the coefficient of restitution decreases as the height increases, subverting the theoretical results initially proposed. I hypothesised that this was due to the air resistance that the ball encounters and that as the height increases, so does the air resistance. For this experiment to be valid, there would need to be only one independent variable but now, there are two (drop height and air resistance).
To make a valid experiment, I would need a formula that integrates air resistance to calculate the velocity of the initial and final bounce height of the ball when dropped, forming the coefficient of restitution.
What can this formula be and how can it be derived from other formulas?

Relevant Equations

K = mg/v2
V = root2gh

Yeah, not sure what to do

 

[haruspex]

Problem Statement: I am conducting an in-class practical investigation where I am investigating whether altering the drop height of a ball will affect the coefficient of restitution.
I have completed all the trials however, the data suggest that the coefficient of restitution decreases as the height increases, subverting the theoretical results initially proposed. I hypothesised that this was due to the air resistance that the ball encounters and that as the height increases, so does the air resistance. For this experiment to be valid, there would need to be only one independent variable but now, there are two (drop height and air resistance).
To make a valid experiment, I would need a formula that integrates air resistance to calculate the velocity of the initial and final bounce height of the ball when dropped, forming the coefficient of restitution.
What can this formula be and how can it be derived from other formulas?
Relevant Equations: K = mg/v2
V = root2gh

Yeah, not sure what to do

It could also be that the coefficient is less at higher impact speeds. See e.g. https://iopscience.iop.org/article/10.1088/1757-899X/36/1/012038/pdf, near the bottom of page 2.

For drag, probably should assume quadratic. Can you write the differential equation for the speed?

 

[NickMitch]

Do you know formula that can integrate air resistance to find velocity?
Possible methods to find that would also help.
Keep in mind, this is year 11 physics

 

[NickMitch]

It could also be that the coefficient is less at higher impact speeds. See e.g. https://iopscience.iop.org/article/10.1088/1757-899X/36/1/012038/pdf, near the bottom of page 2.

For drag, probably should assume quadratic. Can you write the differential equation for the speed?

Possible equation?

 

[haruspex]

Possible equation?

Not sure what you are asking. Are you asking me to provide the equation?
I think you can make an attempt. Suppose the drag force is cv² where v is the velocity and c is some constant. What is the sum of forces on the falling ball? What is the resulting acceleration?