Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

So, I have to say, I am no math wiz as many FIRSTers are, so I was looking for a little friendly help.

If a ball is launched at 45 degrees from a height of 5 feet and the maximum velocity is 12m/s what is the distance traveled when the ball is going down at a height of 8ft 6in? Myself and a friend were able to calculate this WITHOUT taking into account air resistance but were unable to WITH air resistance.

We do however have something to base our model off of. The rules state:

The ball will go approximately 35ft if shot from a 30 degree angle. This manages to take into account neither the 8ft 6in high goal OR the 45 degree best angle but does take into account air resistance.

So, any math wiz' wana' take a crack at it? I am sure MANY teams will be very grateful.

Re: Nerf ball starting at 12m/s 45 degree angle ending at 10ft w/ air resistance
 

I'm not totally sure how it takes into account air resistance, but this applet is a very useful way to quickly and visually model launching things at different speeds and angles.

Re: Nerf ball starting at 12m/s 45 degree angle ending at 10ft w/ air resistance
 

As I said we were able to calculate it without air resistance, but as I am sure you know those nerf balls have a LOT of it.

It is interesting the applet you sent me says it can take into account air resistance yet it does not need surface area? Strange.

I attached a picture of what I need incase anyone is confused.

Re: Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

I might note that 45 degrees is the best angle for range when the starting and ending height are the same, but not when they are different.

Re: Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

Heh. Excellent point. But lets try for 45 degrees and go from their.

Well guys this is sad. Why can't we, the engineering minds of tomorrow, figure this out?

I swear the guys from FIRST know the answer and are laughing their a****s off at us.

If any of you are wondering why this is soo important to your strategy, it means that you need not defend ANY shooter outside this arch, no matter how good.

Re: Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

before you can calculate the answer you requested, someone has to measure the air resistance of the nerf balls.

someone has suggested the first step in another thread: drop one of the balls from a height and measure its terminal velocity (the speed at which its air resistance matches the pull of gravity).

Without real data parameters any calculations will be useless.

Re: Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

Has anyone taken the direction of spin into consideration? I.E. Backspin, frontspin, sidespin

Re: Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

We already have something to base the air friction on, FIRST told us that it will travel 35feet at 30 degrees. That is WITH air resistance. We should be able to base our calculations off that.

The 35ft at 30 degrees is with no spin, so we can do some tests with spin afterwards, but lets get the basics first, i.e. without spin.

Re: Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

Has anyone done weight checks to determine the the population range and median? ;)


[**Time to mess with their heads**]

Re: Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

The problem is that air resistance is highly dependent on SO many factors. Dropping the ball to find its terminal velocity will only give you the drag coefficient at one velocity. It's entirely possibly that the drag coefficient is ridiculously low while the ball is above 10 m/s or so due to turbulence and the roughness of the skin. At any rate, I've still cobbled together a spreadsheet to estimate the range of the ball given air resistance. It should be showing up in the white papers in a few hours.

Re: Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

Done. Using standard quadratic drag estimate and coefficient of .5 (close to a percent or less with spheres traveling less than 33 mph IIRC)
Using matca, someone can check my numbers but I'm pretty sure they're right.
Table is in radians, fired at 12 m/s. The first one is radians, the second entry is the data, 1st entry is the y distance, 2nd entry is the error (from discrete modeling).

{{0.01, {1.878, -0.173984}}, {
0.02, {1.87773, -0.154081}}, {0.03, {1.87729, -0.13419}}, {
0.04, {1.87668, -0.114314}}, {0.05, {1.87589, -0.0944532}}, {
0.06, {1.87492, -0.0746112}}, {0.07, {
1.87377, -0.0547895}}, {0.08, {1.87246, -0.03499}}, {0.09, {
1.87096, -0.0152148}}, {0.1, {2.74991, -0.189958}}, {0.11, {2.74725, \
-0.16047}}, {0.12, {2.74434, -0.131048}}, {
0.13, {2.74117, -0.101694}}, {0.14, {2.73775, -0.0724108}}, {0.15, \
{2.73407, -0.0432013}}, {0.16, {2.73015, -0.0140684}}, {0.17, {3.56738, \
-0.20893}}, {0.18, {3.56169, -0.170432}}, {
0.19, {3.55568, -0.132036}}, {
0.2, {3.54934, -0.0937462}}, {0.21, {3.54268, -0.0555826}}, {
0.22, {3.53569, -0.0175625}}, {0.23, {
4.33218, -0.243732}}, {0.24, {4.32297, -0.196783}}, {0.25, {
4.31338, -0.150017}}, {0.26, {4.3034, -0.103439}}, {0.27, {
4.29303, -0.057053}}, {0.28, {4.28227, -0.0108628}}, {0.29, {
5.03816, -0.270303}}, {0.3, {5.02477, -0.21564}}, {
0.31, {5.01094, -0.161219}}, {0.32, {4.99665, -0.107074}}, {
0.33, {4.98192, -0.0532141}}, {0.34, {5.69995, -0.35631}}, {0.35, {
5.68229, -0.294547}}, {0.36, {5.66411, -0.233119}}, {0.37, {
5.64544, -0.172032}}, {
0.38, {5.62625, -0.11129}}, {0.39, {5.60655, -0.0508989}}, {
0.4, {6.28483, -0.391372}}, {0.41, {6.26187, -0.323656}}, {
0.42, {6.23836, -0.256343}}, {0.43, {
6.21427, -0.189469}}, {0.44, {6.18963, -0.12305}}, {0.45, {
6.16442, -0.0570912}}, {0.46, {6.80323, -0.437435}}, {0.47, {
6.77445, -0.364877}}, {0.48, {6.74505, -0.292836}}, {0.49, {
6.71503, -0.221317}}, {0.5, {6.68439, -0.150323}}, {0.51, {
6.65314, -0.07986}}, {0.52, {6.62127, -0.00993174}}, {0.53, {
7.21763, -0.426246}}, {0.54, {7.18184, -0.350434}}, {0.55, {
7.14538, -0.275235}}, {0.56, {7.10825, -0.200677}}, {0.57, {
7.07044, -0.126763}}, {0.58, {7.03197, -0.0534979}}, {0.59, {
7.58909, -0.515398}}, {0.6, {7.54641, -0.437034}}, {0.61, {
7.503, -0.359378}}, {0.62, {7.45887, -0.282434}}, {0.63, {
7.41401, -0.206206}}, {0.64, {7.36842, -0.130696}}, {0.65, {
7.32211, -0.0559089}}, {0.66, {7.83634, -0.56063}}, {0.67, {
7.78549, -0.481498}}, {0.68, {7.73385, -0.403148}}, {0.69, {
7.68142, -0.325601}}, {0.7, {7.62822, -0.248882}}, {
0.71, {7.57423, -0.172992}}, {0.72, {7.51946, -0.0979347}}, {
0.73, {7.4639, -0.023711}}, {0.74, {7.93061, -0.570386}}, {
0.75, {7.8701, -0.492712}}, {0.76, {7.80875, -0.415932}}, {0.77, \
{7.74656, -0.340046}}, {0.78, {7.68353, -0.265058}}, {
0.79, {7.61965, -0.19097}}, {0.8, {7.55493, -0.117784}}, {0.81, {7.48936, \
-0.0455023}}, {0.82, {7.90715, -0.6387}}, {0.83, {7.83621, -0.563765}}, \
{0.84, {7.76435, -0.489791}}, {0.85, {7.69159, -0.41678}}, {0.86, {7.61792, \
-0.344753}}, {0.87, {7.54334, -0.273721}}, {0.88, {7.46784, -0.203685}}, \
{0.89, {7.39143, -0.134647}}, {0.9, {7.3141, -0.0666068}}, {
0.91, {7.67639, -0.70757}}, {0.92, {7.59316, -0.637761}}, {
0.93, {7.50895, -0.569005}}, {
0.94, {7.42376, -0.501302}}, {0.95, {7.33758, -0.434655}}, {
0.96, {7.25041, -0.369063}}, {0.97, {7.16226, -0.304527}}, {
0.98, {7.07311, -0.241048}}, {0.99, {6.98297, -0.178627}}, {
1., {6.89184, -0.117264}}, {1.01, {6.79971, -0.0569607}}, {
1.02, {7.09238, -0.746587}}, {1.03, {6.99362, -0.685404}}, {1.04, \
{6.89378, -0.625334}}, {1.05, {
6.79287, -0.566376}}, {1.06, {6.69088, -0.508531}}, {1.07, {
6.58781, -0.4518}}, {1.08, {6.48366, -0.396184}}, {1.09, {6.37843, \
-0.341692}}, {1.1, {6.27211, -0.28833}}, {1.11, {6.1647, -0.236096}}, {1.12, {
6.0562, -0.184991}}, {1.13, {5.9466, -0.135014}}, {1.14, {5.83591, \
-0.086165}}, {1.15, {5.72412, -0.0384431}}, {1.16, {5.9213, -0.780452}}, \
{1.17, {5.80176, -0.732864}}, {1.18, {5.68103, -0.686452}}, {1.19, {
5.55911, -0.641218}}, {1.2, {5.43599, -0.597161}}, {1.21, {5.31167, \
-0.55428}}, {1.22, {5.18615, -0.512575}}, {1.23, {5.05942, -0.472047}}, \
{1.24, {4.93149, -0.432694}}, {1.25, {4.80234, -0.394516}}, {1.26, {4.67198, \
-0.357514}}, {1.27, {4.5404, -0.321686}}, {1.28, {4.4076, -0.287034}}, {1.29, \
{4.27357, -0.253555}}, {1.3, {4.13832, -0.221251}}, {1.31, {4.00184, \
-0.19012}}, {1.32, {3.86413, -0.160163}}, {1.33, {3.72518, -0.13138}}, {
1.34, {3.58499, -0.103769}}, {1.35, {3.44356, -0.0773314}}, {1.36, \
{3.30088, -0.0520663}}, {1.37, {3.15696, -0.0279736}}, {1.38, {3.01179, \
-0.00505312}}, {1.39, {3.02363, -0.810602}}, {1.4, {
2.86829, -0.789156}}, {1.41, {2.71157, -0.76893}}, {1.42, {
2.55347, -0.749925}}, {1.43, {2.39399, -0.732141}}, {1.44, {
2.23313, -0.715577}}, {1.45, {2.07088, -0.700233}}, {1.46, {1.90724, \
-0.686109}}, {1.47, {1.7422, -0.673205}}, {1.48, {1.57576, -0.661522}}, \
{1.49, {1.40792, -0.651058}}, {1.5, {
1.23868, -0.641814}}, {1.51, {1.06802, -0.633789}}, {1.52, {
0.895955, -0.626985}}, {1.53, {0.722471, -0.6214}}, {1.54, {
0.54757, -0.617034}}, {1.55, {0.371248, -0.613888}}, {1.56, {0.193502, \
-0.611962}}, {1.57, {0.0143294, -0.611255}}}

If anyone wants me to generate some other arbitrary thing (rather than firefrom x = 0 to x =0 .. I can do so.)

Re: Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

can someone tell me if these are real numbers lol...

looking through them they look right but cha never no oy =/...

Re: Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

Oops the drag coefficient should be .416 not .5 .. the difference should be small, but I guess if you're picky, I'll go regenerate the table..

for anyone wanting to check my constants

.416 = Cd
1.29 kg/m^3 = rho
3.5^2 Pi = area (convert to meters)

so we get ..
Fdrag = v^2 * 0.00680265...

that wasn't so bad eh?
now just do some discrete modeling assuming a is constant over small intervals (deltat is .1 in this example.. smaller would be more accurate, but this is a rough estimate anyway, density isnt contant and spin effects would negate this..)

*edit* - I /shoud/ be cramming for my finals, but if I finish my final code, I'll paste the ballistics table here with variables v, vbot, theta... It shouldn't be that bad, once I'm done with that... I can take into account lateral air resistance (the list keeps going). I'll post them here when I'm done (hopefully after friday, my physics final =p )

*edit 2* - is someone with matca willing to lend some processing time? I realized that a delta t < .1 (ie .001) takes a ridiculously long time... if anyone is willing to help generate ballistics table, I'd be grateful..

Tatsu

Re: Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

Sorry Tatsu but that math is WAY to advanced for me.

However if you need some lended processing power I will do what ever you ask, but you are going to have to give some pretty clear instructions.

Re: Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

Not sure what I'd need.. I just need someone with Mathematica 5.1 that can execute my notebook.. if not I'll just have to run it overnight or something.. not that big of a problem.

Tatsu.

Anyway, the equation isn't that bad if you dont try to make it a diffeq problem.. its just small timestep modeling.. and yes, you probably cant do this by hand (and thus the RC wouldnt be able to either.)

so if you want accurate trajectories, you'll need lookup tables.. (I cant think of any other way to calculate air resistance trajectories on the fly..)

Re: Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

Just curious.. where did you find this?

Re: Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

Taken from Section 4 - The Game

<S02> Muzzle Velocity - No ROBOT may throw a ball with an exit velocity of greater than 12 m/s (26.8 mph).
As a reference, a ball traveling at this velocity when leaving the ROBOT at an angle of 30º from
horizontal with no spin will travel approximately 35 feet. A robot that violates this rule will be
considered unsafe per <S01>.

Re: Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

And the mean, mode and standard deviation? I also want to see a scatter plot to check to see if it is a one-tail or two-tail population, or even one with more than one mode. Some sort of 3D modelling of mass density would be nice, too.

Re: Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

I massed 6, using a mean of .18 ... the mass deviations dont significantly affect the range..

Re: Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

When I posted this all I wanted was that distance. So we can make a "no one can shoot past here arc" on our field.

Re: Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

oh, is THAT what you want?

I think that's 6.5 meters. well, I'll just generate the full table and find the max, faster that way =p.

*edit* - wow, finals studying is making me drowsy.. but the max should be around 8.8 meters.. give or take maybe .5 meters.. (sorry, I dont have a full lookup table, but this seems pretty close..)

*edit 2* - does anyone still want the lookup tables?

Re: Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

I might if I knew what they were... all I know that I need is that one length, what will that table give me?

Also, is there any way to calculate what the best angle would be instead of 45 degrees?

Re: Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

Well, a standard lookuptable is something like this..

y axis - theta
x axis - v

the entry for some v and theta gives you the distance (in this case from 5 ft to 81/2 ft).. therefore you reverse-lookup to find a theta. Lets say you keep v constant at 12 m/s Now just find an entry in the table which is equal do your distance from the goal... that gives you your fireing angle. Fire.

Otherwise you have to calculate angle depending on your distance, that's not possible when you factor air resistance.. (at least not to my knowelege).. so you have 2 options

a). use lookup tables to find an angle that matches your distance

or b). use a model without air-resistance and be off by a large amount the farther away you get.

Either way its not pretty, but I think our team wants to do a). (psst.. dont tell them I already made the lookup table, I want to see how fast our programmers get it =p )

Re: Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

LoL! So mean... so mean....

Re: Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

Here is a picture that show a theoretical max distance. It ignores the effects of drag and spin.
It shows the theoretical max distance, when the ball is launched from ground level, at 12m/s and 45 deg. elevation to be about 37.2 ft/11.34m.

Re: Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

In another thread, I posted a link to a paper that describes how to calculate the trajectory of a ball, including air resistance. I also attached an Excel file that allows the user to vary launch speed, angle and height as well as ball mass, diameter, drag coefficent. It produces a trajectory plot with the upper and lower heights for the goal shown on the graph. It's pretty easy to use and gives you a tool to determine how sensitive the trajectory is to the various parameters.

See:
http://www.chiefdelphi.com/forums/sh...710#post429710

Re: Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

Im no math whiz, but would this work? have your launcher (for instance turret like a snowblower) and mount the cam so its on the top of the launcher, then proportionally program the offset of cam-turret to greenlight-hole? that way when your bot drives the cam always tracks the greenlight and your launcher is always aimed there. Me=mechanical and not too electrically intuitive, is this possible?

Re: Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

By the way billbo911's calculations were provided by:
www.hyperphysics.phy-astr.gsu.edu

We got:
From 1.4 meters off ground to center of goal, we calculated that max range 11.1 meters at 48 degrees. That includes drag coefficient of 0.3 - constant drag coefficient. The proof is left for an exercise for the reader. :D

Re: Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

How'd you get that number? you're off about 3 meters from my calculations.. weird. want to run your caclulations again with a coefficient of .416 (should be more accurate)? I might be wrong..

Re: Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

Air resistance is not a factor unless you are going 75+ mph. So you dont have to account air resistance into the equation.

Re: Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

Aerodynamic drag is definitely a factor for a fluffy object like a Poof ball. It's not an extremely large factor, but it has a real and noticeable effect.

Re: Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

You're kidding right? You lose around 4 meters to air resistance fireing 45 degrees from 0 -> 0 at 12 m/s. This is fairly nontrivial. Initial deceleration is easy to calculate

F= 1/2 Cd * rho * A * v^2

for a poof ball, thats's about

F= 7.52e-3 * v^2

F= 1.08 Newtons.. that may seem insignificant.. but the poof ball is light..

F=ma=1.08 , so taking m = .18 kg we get a = 6.016 m/s^2 deceleration..

yes, that's about 2/3rds g, that's nontrivial.

Re: Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

You drive a Honda Scion, dont you?

(Aerodynamics ?! we dont need no stinking aerodynamics! :^)

Re: Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

ugliest...vehicle...ever...

Re: Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

Btw, the 2-dimentional version of calculation is done.

check the programming section for my mathematica notebook its called something like traj.zip.. gimmie a day and I'll also upload the raw data in CSV form.

Re: Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

For launching 4.5 feet to 8.5 feet:

Please input the initial velocity in meters per second: 12.8
Please input the coefficient of drag (0.07 - 0.5): 0.416
The uber best range is 14.7775 meters at an uber angle of 47 degrees.


C++ Code:
#include <iostream.h>
#include <math.h>

int main()
{
// float i = cos(x); x is in radians
// float i = atan(x); i is in radians arctangent

float x;
float y;
float v_initial = 12.8;
float v_x;
float v_y;
float initial_angle;
float theta;
float drag;
float area = pow((3.5 * 2.54 / 100), 2) * 3.14159;
float mass = .206; //kilograms
float C_D; //Coefficient of Drag
float a_x;
float a_y;
float increment = .01; //Incrementing time
float best_range;
float rad_conv = 2.0 * 3.14159 / 360.0;
float uber_best_range = 0;
float uber_angle = 0;

cout << "This program will calculate the maximum range and best angle to launch a 7 inch diameter poof ball." << endl << endl;
cout << "Please input the initial velocity in meters per second: ";
cin >> v_initial;

cout << "Please input the coefficient of drag (0.07 - 0.5): ";
cin >> C_D;

for(initial_angle = 10; initial_angle <= 80; initial_angle++)
{
x = 0;
y = 0;
v_x = v_initial * cos(initial_angle * rad_conv);
v_y = v_initial * sin(initial_angle* rad_conv);
best_range = 0;

do
{
theta = atan(v_y/v_x);
drag = .5 * C_D * area * sqrt(pow(v_x, 2) + pow(v_y, 2));
a_x = drag / mass * cos(theta);
a_y = drag / mass * sin(theta) + 9.8;

v_x = v_x - a_x * increment;
v_y = v_y - a_y * increment;

x = x + v_x * increment;
y = y + v_y * increment;

if(y > 1.3 && v_y < 0)
best_range = x;

}while(y >= 0);

cout << "At " << initial_angle << " degrees, the best range is " << best_range << " meters." << endl << endl;

if(best_range > uber_best_range)
{
uber_best_range = best_range;
uber_angle = initial_angle;
}

}

cout << "The uber best range is " << uber_best_range << " meters at an uber angle of " << uber_angle << " degrees.";

return 0;
}

--Ownage

Re: Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

nice, nearly identical code, different language.

(btw, you might want to run it for <1 degree increments)

Re: Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

Have I been trolled?

The Scion Xb is Toyota's ugly cube. Honda's ugly cube is the Element.

Re: Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

971 got .41 ish as our Coefficient of drag (using wind-tunnel testing). Can we assume this is verified. Also, I don't know if this has been explained, but the "rho" in the drag equation is air density, generally around 1.2 kg/m^3.

Re: Nerf ball starting at 12m/s 45 degree angle ending at 8ft 6in w/ air resistance
 

.41 matches prediction. prediction is .416 with flow of 12m/s