Projectile motion with air resistance?

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I'm trying to calculate projectile motion while taking air resistance into account. All my physics teacher could tell me was that it involved differential equations.

Could anyone help me out with this? Perhaps point to a site that explains it...

all I know is it's usually a parabola of some sort.

yeah, that's how it works without air resistance... well, I suppose it would still be parabolic with air resistance. I'll probably be making a few simplifying assumptions along the way. It's for a calculus project and I have to calculate shooting something halfway across the country.

the air resistance is always acting in the exact opposite direction than path.
it gets higher with speed, by a factor of 2nd power.
the higher it gets, the more it slows the projectile, which reduce it's impact on speed (that's where you need calculus).

hope that helps a bit.... hope you're not a terrorist building a missile ?

Try this:



(because of the backgrounding it may be kind of hard to read - click on it to see the original)

This is a 2nd order nonlinear differential equation in 3 variables (or one 3D vector) for the (approximate) position of a free sphere with radius D (in meters), with mass m (in kg), close enough to earth where g is a good approximation of the acceleration due to gravity in a vacuum, in regular air at STP (this info quoted from Classical Mechanics, by Taylor, pp 71-72)

Note that it is second order in a 3D vector, just like newton's 2nd law (recognize the "F=ma=mg" part)

Also note that it's nonlinear, due to the extra dependence on the magnitude of the velocity of the sphere in the quadratic term.

Thirdly note that beta is significantly smaller than gamma. Therefore, the quadratic (and nonlinear) component of the air friction dominates the linear component (which is also linear in the sense of the diffeq, no coincidence). Therefore, you cannot take the linear diffeq you get from dropping the quadratic term as a good approximation.

I forget if closed form solutions to this exist (I just grabbed the book, I only skimmed through it for the equation), but usually, with nonlinear stuff people head straight to the computer to solve it numerically.

edit: and fourthly note that coyote is right - it is primarily quadratic, the frictional force is always opposite to the velocity, and that also, in general, the path is not a parabola or any other conic section of any sort.

Thanks guys, I'll take a look at what you posted and see if I can go from there. I'm in high school, so this is a bit past what I'm used to.

No, but for an open-ended project in BC Calculus my partner (assigned) insisted that we calculate what it would take to fire our textbook halfway across the country to hit its author at the university he teaches at...(sigh) I wanted to go ahead and throw in the air resistance, so we were actually doing CALCULUS, but she didn't know how, which is why I'm now asking for help here.

I like the idea
I guess that if you could really do it, you would literally fire it (it would ignite because of the enormous speed).

You might try a numerical method, a la Euler's method. Don't know where you are in calculus, but something like:

dt = time-slice, some small value
vx = initial x-velocity
vy = initial y-velocity
g = grav. accelleration
a = coeff. of air resistance

x = y = t = 0

while ( y >= 0 ) { // stop when the projectile hits the ground

Fx = -a * ( vx^2 + vy^2 ) * sign(vx)
Fy = -g * m - a * ( vx^2 + vy^2 ) * sign(vy)

ax = Fx / m
ay = Fy / m

vx = vx + ax * dt
vy = vy + ay * dt

x = x + vx * dt
y = y + vy * dt

t = t + dt

plot (x,y) // or whatever
}


You could tweak Fx and Fy to suit what model of air resistance you want, and if you want to get fancy use the Runga-Kutta (sp?) or some other method.

Unless you really know what you're doing, implementing your own numerical diffeq solver is not really easy at all - there are so many little subtleties that tend to crop up.

If you have access to mathematica or matlab or such, things are much easier - just use their built in numerical integrators. (for example, see: http://reference.wolfram.com/mathematica/tutorial/NDSolveIntroductoryTutorial.html )

Well this problem seems simple enough -- it's not like it's a fluid dynamics problem or something. I did this sort of thing in high school constantly and it worked well enough (and later learned in college what it was). Estimating/controlling the error might be tough, but I guess that would depend on how rigorously the instructor wants things done.

Yeah, you're probably right. I'm just averse to implementing numerical algorithms, I think

How would I calculate it if I don't have access to mathematica, maple etc?

It depends what your "etc" excludes.

If you have access to a Linux system, I guess Maxima will suffice.

In any case, a program (in C, Basic, whatever) as outlined by PlainBlueSky will work. You would need to"hunt around" with initial launch speed and angle to see what would give the range you require. Also, retrying the solution with varying time steps will be needed to give confidence in its accuracy.

And... the short answer to "How would I calculate it" is that you can't. The best you will manage is a numerical approximation.

Drop air resistance, and you'll have less hassle, except that the range you are after is still going to give you a problem, as it is not a parabolic trajectory, even to first order, so you should be treating it as an ellipse and solving it as a problem in orbital mechanics.

If you keep the air, I'd guess that the solution is that it is impossible. As korppi remarked earlier, the velocity needed to attain the range will certainly result in anything made of paper (and most other materials) instantly vaporizing.

If your "books" consisted of nanotech encodings on a pinhead at the centre of a large ceramic ball, maybe you could get the range.

Maxima is available for Windows. And I used to do all my electronic calculations from simple Ohm's law in DC circuits to
calculating RCL impedances using a Quattro-Pro spread-sheet program back in the nineties...

I'm on a Mac. I'm trying to see if I can get access to Mathematica, but I'm not too confident about it.

Mathematica costs money. Why bother?

Also, spreadsheets are pretty good for this sort of thing. I once wrote a helicopter simulation in a spreadsheet.