Why Does My Ball Trajectory Simulation Show Only Positive Vertical Velocities?

[Chelonian]

I am attempting to create a spreadsheet that models the flight path of a sphere. My model should consider air resistance, but I've opted to ignore magnum. As a resource to help me, I'm using http://www.team2834.com/team_documents/Projectile_motion_with_air_resistance.pdf, but I have ran into some trouble. I have the velocity and position defined recursively as:
$$v_x(n+1)= v_x(n)+a_x(n)Δt$$
$$v_y(n+1)= v_y(n)+a_y(n)Δt$$
$$x(n+1)=x(n)+v_x(n)Δt+a_x(n)(Δt)^2$$
$$y(n+1)=y(n)+v_y(n)Δt+a_y(n)(Δt)^2$$
The acceleration has given me some trouble, though. I tried defining acceleration as it the page seemed to indicate:
$$a_x(n)=-(D/m)v_x(n) \sqrt{{v_x(n)}^2+{v_y(n)}^2}$$
$$a_y(n)=-g-(D/m)v_y(n) \sqrt{{v_x(n)}^2+{v_y(n)}^2}$$
When I used this to finish my spreadsheet, I noticed that the vertical velocity was always positive, an obvious error. It is very possible that I have made an error unrelated to the acceleration equations (I am a novice at best at mechanics and Excel), but I saw this as the most likely candidate. The [erroneous] spreadsheet I currently have is attached.

What is causing these incorrect values? If there is any additional information I can give to help, let me know, and I will be happy to do so. I'd hate for all of my work to be wasted, so any help you can give is very much appreciated. Thank you for your time and for your assistance.

 

[voko]

Your vertical acceleration does not include the term due to gravity. This is seen directly from the numeric values: its magnitude should always be greater than 9.8 while the vertical velocity is positive, and always less than 9.81 otherwise.

I have also noticed that you inserted some ad hoc calculations of the position and vertical velocity in the first iteration. When I removed those, and added the missing gravity term to vertical acceleration, I got negative vertical velocity on the 7th iteration, and negative altitude at the 16th iteration.

 

[Chelonian]

Your vertical acceleration does not include the term due to gravity. This is seen directly from the numeric values: its magnitude should always be greater than 9.8 while the vertical velocity is positive, and always less than 9.81 otherwise.

I have also noticed that you inserted some ad hoc calculations of the position and vertical velocity in the first iteration. When I removed those, and added the missing gravity term to vertical acceleration, I got negative vertical velocity on the 7th iteration, and negative altitude at the 16th iteration.

Thank you very much. I don't know how my g value slipped out of my functions, but I'm glad I fixed it. When I just made that change, I ended up with negative values one iteration earlier then you did. What should the initial position and velocity equations be? I don't see what's wrong with them.
To clarify, is the acceleration formula in my first post correct, or should it be $a_x(n+1)$ and $a_y(n+1)$ instead of $a_x(n)$ and $a_y(n)$?
I made a new version of the spreadsheet which I think addresses and fixes the problems you mentioned. Let me know if there's anything I need to fix on this one.
Thanks again for the help.

 

[voko]

It looks good to me now. The only thing I would change is the time step, I would make it much smaller, you do not want big velocity changes between steps.

 

[PJC01]

Hello,

Thank you for reaching out for help with your ball trajectory simulation project. It seems like you have put a lot of effort into this and I am happy to assist you in troubleshooting the issue you are experiencing.

Based on the information and equations you have provided, it appears that your approach to modeling the ball's trajectory is correct. However, I noticed that you are using the same equations for both horizontal and vertical acceleration. This could be the reason for the incorrect values in your spreadsheet.

The correct equations for acceleration should be:

$$a_x(n)=-(D/m)v_x(n) \sqrt{{v_x(n)}^2+{v_y(n)}^2}$$
$$a_y(n)=-g-(D/m)v_y(n) \sqrt{{v_x(n)}^2+{v_y(n)}^2}$$

As you can see, the horizontal and vertical accelerations have different components. The horizontal acceleration is affected by the drag force, while the vertical acceleration is affected by both the drag force and the force of gravity.

I recommend double-checking your equations and making sure that you have the correct values plugged in for all variables. It is also helpful to break down the equations into individual steps to make sure you are getting the correct values for each variable.

If you are still experiencing issues with your spreadsheet, please feel free to provide more information or share your spreadsheet with me so I can take a closer look. I am happy to help you troubleshoot further.

I wish you the best of luck with your project and thank you for your dedication to understanding projectile motion with air resistance. It is a complex topic, but with persistence and careful analysis, I am confident you will achieve your desired results.

Best,