# Ball Trajectory Simulation

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 this page, 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.

#### Attachments

• Ball Trajectory Simulator.xlsx
18.4 KB · Views: 136

## Answers and Replies

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.

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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.

#### Attachments

• Ball Trajectory Simulator - 1.xlsx
12.8 KB · Views: 230
Last edited:
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.