paper: Parabolic Trajectory Calculations

The definition of terminal velocity in the context of this thread is given in the first paragraph of this web page:

You are using a different definition.

Models with air drag are “Newtonian” too. The acceleration is still equal to the net force divided by the mass (Newton’s 2nd law).

Perhaps what you meant is constant acceleration model equations ignore air drag.

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[b]Parabolic vs Air-Drag Trajectory revC

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Parabolic vs Air Drag Trajectory revC is the same as revB except the graph is not auto-scaling. Some folks may prefer this.

Parabolic vs Air Drag Trajectory revB fixes a small error: the “launch height” user input parameter was not being imported into the parabola equation.

The physics and math for the computer numerical simulation (including topspin/backspin)
is explained in this paper.

Parabolic Equations and constants a, b, c, xp, and yp
y = ax[sup]2[/sup] + bx + c
and y = a*(x-xp)[sup]2[/sup] + yp explained:

How to find constants a and b, given launch speed and angle:

How to find constants a and b, given desired scoring range:

How to find constants a and b, given yp and a point (x1,y1) on the trajectory:

parabola.pdf (48.2 KB)

compute a&b from yp,x1,y1 rev02.pdf (16.6 KB)
parabola vs air drag.zip (57.5 KB)
parabola vs air drag revB.zip (51.7 KB)
parabola vs air drag revC.zip (63.6 KB)
Terminal Velocity.zip (26.5 KB)
given Vo and (d,h) find theta.pdf (44.6 KB)

I’ve recently received a couple of PMs asking about the formulas in the spreadsheet. I’m going to post summaries of the answers here so interested students may benefit:

Some cells are hidden to make the user interface cleaner. To make those cells visible: Unhide columns GHIJK, move the graph out of the way, and highlight the whole spreadsheet.

The air drag acceleration vector D always points 180 degrees opposite to the Velocity vector V.

The magnitude of D is modeled as:

(V2/V[sub]t[/sub]2)*g

… where V[sub]t[/sub] is the magnitude of the terminal velocity.

The magnitude of the X component of D is

(V2/V[sub]t[/sub]2)gcos(θ)

= (V2/V[sub]t[/sub]2)g(Vx/V)

= (V*V[sub]x[/sub]/V[sub]t[/sub]2)*g

The magnitude of the Y component of D is

(V*V[sub]y[/sub]/V[sub]t[/sub]2)*g