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Parabolic and Air-Drag Trajectory Calculations

Ether

By: Ether
New: 02-20-2014 07:49 PM
Updated: 02-17-2016 06:11 PM
Total downloads: 3810 times





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 = a*x2 + b*x + c
and y = a*(x-xp)2 + yp explained:
http://www.chiefdelphi.com/media/papers/download/3868

How to find constants a and b, given launch speed and angle:
http://www.chiefdelphi.com/media/papers/download/3871

How to find constants a and b, given desired scoring range:
http://www.chiefdelphi.com/media/papers/download/3872

How to find constants a and b, given yp and a point (x1,y1) on the trajectory:
http://www.chiefdelphi.com/media/papers/download/3877

Terminal Velocity calculator spreadsheet:
http://www.chiefdelphi.com/media/papers/download/3900




Attached Files

  • pdf Introduction

    parabola.pdf

    downloaddownload file

    uploaded: 02-20-2014 07:49 PM
    filetype: pdf
    filesize: 48.16kb
    downloads: 389


  • gif Compute a&b from speed&angle

    given speed&angle, find a&b.gif

    downloaddownload file

    uploaded: 02-20-2014 09:21 PM
    filetype: gif
    filesize: 11.06kb
    downloads: 496


  • gif Compute a&b from scoring range

    scoring_range.gif

    downloaddownload file

    uploaded: 02-20-2014 09:30 PM
    filetype: gif
    filesize: 12.08kb
    downloads: 313


  • pdf Compute a&b from yp,x1,y1

    compute a&b from yp,x1,y1 rev02.pdf

    downloaddownload file

    uploaded: 02-21-2014 10:21 AM
    filetype: pdf
    filesize: 16.63kb
    downloads: 366


  • zip Parabolic vs Air Drag Trajectory

    parabola vs air drag.zip

    downloaddownload file

    uploaded: 02-24-2014 10:46 PM
    filetype: zip
    filesize: 57.46kb
    downloads: 175


  • zip Parabolic vs Air Drag Trajectory revB

    parabola vs air drag revB.zip

    downloaddownload file

    uploaded: 02-27-2014 05:17 PM
    filetype: zip
    filesize: 51.67kb
    downloads: 331


  • zip Parabolic vs Air Drag Trajectory revC

    parabola vs air drag revC.zip

    downloaddownload file

    uploaded: 03-04-2014 12:28 PM
    filetype: zip
    filesize: 63.61kb
    downloads: 1194


  • zip Terminal Velocity spreadsheet

    Terminal Velocity.zip

    downloaddownload file

    uploaded: 03-04-2014 01:02 PM
    filetype: zip
    filesize: 26.53kb
    downloads: 208


  • pdf Given launch speed and a point on the parabola, find theta

    given Vo and (d,h) find theta.pdf

    downloaddownload file

    uploaded: 02-17-2016 06:11 PM
    filetype: pdf
    filesize: 44.59kb
    downloads: 336



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02-27-2014 05:23 PM

Ether


Unread Re: paper: Parabolic Trajectory Calculations


For those of you who downloaded the Parabolic vs Air Drag Trajectory spreadsheet, please note that I just uploaded revB to correct a small error. The user input "launch height" was not being imported into the parabola equation.



02-28-2014 11:07 AM

Ether


Unread Re: paper: Parabolic Trajectory Calculations

Quote:
Originally Posted by Ether View Post
For those of you who downloaded the Parabolic vs Air Drag Trajectory spreadsheet, please note that I just uploaded revB to correct a small error. The user input "launch height" was not being imported into the parabola equation.
I should have mentioned:
The parabola plot with the original version was correct as long as you didn't change the launch height.

And even if you did change the launch height, the error affected only the parabola, not the air-drag trajectory.




03-04-2014 08:55 AM

Hugh Meyer


Unread Re: paper: Parabolic Trajectory Calculations

Ether,

Would you more clearly define the launch angle? A drawing would be nice. Is that from the horizon, or a plumb line? Thanks.

-Hugh



03-04-2014 09:19 AM

Ether


Unread Re: paper: Parabolic Trajectory Calculations

Quote:
Originally Posted by Hugh Meyer View Post
Would you more clearly define the launch angle? Is that from the horizon, or a plumb line?
It's the elevation angle (from the horizontal).

The reason you don't see the graph appear to visually correspond to the launch angle is because the X and Y axes are not scaled equally, and when you change the launch angle the scaling auto-adjusts to fit the graph.

While you're here, does your team happen to have any test data to confirm (or refute) the 37 ft/sec terminal velocity number for this year's game piece?



03-04-2014 09:36 AM

Hugh Meyer


Unread Re: paper: Parabolic Trajectory Calculations

Thank you.

We do not have any data, but we have been talking about it. How would we measure the terminal velocity?

-Hugh



03-04-2014 11:23 AM

marccenter


Unread Re: paper: Parabolic Trajectory Calculations

Ether,

The assumption appears reasonable with the shooting range we are seeing on our robot.

I just found the thread today. When we unbag the robot on Saturday, I will be attempting to increase our shooting percentage via angle and may be able to provide some data after that point.

We did overshoot the target a bit last weekend at Southfield, MI a few more times than I would have liked.



03-04-2014 11:28 AM

Ether


Unread Re: paper: Parabolic Trajectory Calculations

Quote:
Originally Posted by marccenter View Post
The assumption appears reasonable with the shooting range we are seeing on our robot.
I think there might be a misunderstanding. Terminal Velocity is independent of shooting range. It is a function of mg, Cd, rho, and A only.




03-04-2014 01:00 PM

Ether


Unread Re: paper: Parabolic Trajectory Calculations

Quote:
Originally Posted by Hugh Meyer View Post
How would we measure the terminal velocity?
I don't know for sure; I've never done it. Perhaps drop the ball from a sufficient height next to a marked wall in a tall room and take high speed video with a camera that timestamps the frames. Then tweak the value of Terminal Velocity in this spreadsheet until the model matches your data.

I just posted a small revision (revC) to the air-drag spreadsheet. I turned off the auto-scaling in the graph and re-shaped it so the launch angle "looks" more like the real thing. It may make it easier to visualize what's changing when you change the input parameters. The downside is you lose some resolution.

http://www.chiefdelphi.com/media/papers/2946




02-17-2016 06:15 PM

Ether


Unread Re: paper: Parabolic Trajectory Calculations


Given launch speed and a desired point (d,h) on the trajectory, show the derivation of and formulas for the launch angles and the equations of the two parabolic (no air drag) solutions.

http://www.chiefdelphi.com/media/papers/download/4614




02-17-2016 10:44 PM

GeeTwo


Unread Re: paper: Parabolic Trajectory Calculations

Quote:
Originally Posted by Hugh Meyer View Post
How would we measure the terminal velocity?
My first thought is a football stadium and a radar or ultrasonic speed gun. The procedure is pretty obvious.

If you don't have a speed gun, a strobe of some sort, including the video method suggested by Ether would be next.

As for 3946, we did the air-resistance-free calculation, added about 50%, tested that we had more than we needed to hit the goal at the ranges we wanted, then we'll back down based on empirical launch data until we hit the goal at the desired range (this year, with our rear bumper in the outer works). Not as elegant as the full-physics solution, but we've built several high-percentage launchers using this paradigm.



01-11-2017 03:06 PM

Jacob Plicque


Unread Re: paper: Parabolic Trajectory Calculations

Ether
What coeficient of drag are you using for the ball in the 2014 spreadsheet? In 2017 the ball has a Cd in the 0.6 to 0.8 range by looking at wiffle ball data which varies with the Reyonds number.



01-11-2017 03:25 PM

Ether


Unread Re: paper: Parabolic Trajectory Calculations

Quote:
Originally Posted by Jacob Plicque View Post
Ether
What coeficient of drag are you using for the ball in the 2014 spreadsheet?
The calculation is based on the terminal velocity (Cell A5), and an assumption that air drag varies with the square of velocity.





01-13-2017 01:49 PM

peronis


Unread Re: paper: Parabolic Trajectory Calculations

does anyone now the terminal velocity of the fuel?



01-13-2017 03:29 PM

Jacob Plicque


Unread Re: paper: Parabolic Trajectory Calculations

The terminal velocity can be calculated from the peak height and the gravity constant.



01-13-2017 04:23 PM

GeeTwo


Unread Re: paper: Parabolic Trajectory Calculations

Quote:
Originally Posted by peronis View Post
does anyone now the terminal velocity of the fuel?
Search is your friend.



01-13-2017 04:43 PM

Ether


Unread Re: paper: Parabolic Trajectory Calculations

Quote:
Originally Posted by Jacob Plicque View Post
The terminal velocity can be calculated from the peak height and the gravity constant.
OK, so given g=-9.8 meters/sec2 and peak height = 3 meters, please show us how you would calculate the terminal velocity... without using any other information such as launch speed or launch angle.




01-14-2017 08:57 AM

Jacob Plicque


Unread Re: paper: Parabolic Trajectory Calculations

From a known height of 3 meters the calculation follows a freefall model (Vy=0 @t=0) which uses Vy=gt and Y=0.5gt^2.
Thus t=sqrt(3m/9.8m/sec^2/0.5sec)=0.78 seconds.
Vy terminal=0.78sec*9.8m/sec^2=7.66m/sec
The Newtonian trajectory equations do use the initial velocity Voy as follows:
Vy=Voy-gt
Y=Voyt-0.5gt^2 @ Vy=0 the arc is at its peak
X=Voht
Voy=Vo*sin(launch angle from horizon)
Vox=Vo*cos(launch angle)



01-14-2017 09:34 AM

Ether


Unread Re: paper: Parabolic Trajectory Calculations

Quote:
Originally Posted by Jacob Plicque View Post
From a known height of 3 meters the calculation follows a freefall model (Vy=0 @t=0) which uses Vy=gt and Y=0.5gt^2.
Thus t=sqrt(3m/9.8m/sec^2/0.5sec)=0.78 seconds.
You are misunderstanding the meaning of terminal velocity.




01-14-2017 02:51 PM

Jacob Plicque


Unread Re: paper: Parabolic Trajectory Calculations

The Newtonian model equations ignore drag which calculates a velocity only on the basis of gravity. My answer is only valid for a drag free Newtonian calculation. Terminal velocity with drag provide an upper velocity limit for a falling object. For a falling baseball it would be about 33m/sec (~108ft/sec) while a hail stone is ~14m/sec (45ft/sec). A 3 meter drop reaches a peak velocity of 7.66m/sec (~26ft/sec) which is less than the maximum terminal velocity. of either of the above examples.



01-14-2017 03:02 PM

Ether


Unread Re: paper: Parabolic Trajectory Calculations

Quote:
Originally Posted by Jacob Plicque View Post
The Newtonian model equations ignore drag which calculates a velocity only on the basis of gravity. My answer is only valid for a drag free Newtonian calculation. Terminal velocity with drag provide an upper velocity limit for a falling object. For a falling baseball it would be about 33m/sec (~108ft/sec) while a hail stone is ~14m/sec (45ft/sec). A 3 meter drop reaches a peak velocity of 7.66m/sec (~26ft/sec) which is less than the maximum terminal velocity. of either of the above examples.
The definition of terminal velocity in the context of this thread is given in the first paragraph of this web page:

https://en.wikipedia.org/wiki/Terminal_velocity

Please read it.

You are using a different definition.





01-17-2017 06:13 PM

Ether


Unread Re: paper: Parabolic Trajectory Calculations

Quote:
Originally Posted by Jacob Plicque View Post
The Newtonian model equations ignore drag which calculates a velocity only on the basis of gravity.
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.




01-19-2017 08:45 PM

Ether


Unread Re: paper: Parabolic Trajectory Calculations

Quote:
Originally Posted by Ether View Post
Thread created automatically to discuss a document in CD-Media.

Parabolic Trajectory Calculations by Ether


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/Vt2)*g

... where Vt is the magnitude of the terminal velocity.


The magnitude of the X component of D is

(V2/Vt2)*g*cos(θ)

= (V2/Vt2)*g*(Vx/V)

= (V*Vx/Vt2)*g


The magnitude of the Y component of D is

(V*Vy/Vt2)*g




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