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

[Ethulin]

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.

[Greg Marra]

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.

[Ethulin]

Originally Posted by Greg Marra

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.

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.

[Rickertsen2]

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.

[Ethulin]

Originally Posted by Rickertsen2

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.

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.

[KenWittlief]

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.

[Jeff Rodriguez]

Originally Posted by KenWittlief

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.

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

[Ethulin]

Originally Posted by KenWittlief

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.

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.

Originally Posted by Ogre

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

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.

[Bill Moore]

Originally Posted by Ogre

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

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


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

[Kevin Sevcik]

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.

[Tatsu]

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.)

[Heretic121]

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

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

[Tatsu]

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

[Ethulin]

Originally Posted by Tatsu

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

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.

[Tatsu]

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..)