Projectile Motion w/ Drag... help?

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FiveFinger
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Projectile Motion w/ Drag... help?

Postby FiveFinger » Tue May 19, 2009 1:08 pm UTC

(Just a bit of stage setting- I am a high school student, but this is a personal inquiry and the physics teacher is out for a week, and my google-fu has failed miserably.)

Well, obviously, w/o drag, the equations are simple.

Vertical
d(t) = vit + 1/2gt2
v(t) = vi + gt
a(t) = g

Horizontal
d(t) = vit
v(t) = vi
a(t) = 0

But that's really simplified- that assumes an unchanging force of gravity. But for the moment, I just want to start including the effects of drag.

k(vwind - vobject)2 = Fwind = ma = a (assuming mass = 1)

Vertical
a(t) = g + k(vwind - vobject)2

Horizontal
a(t) = k(vwind - vobject)2

That's where I get stuck... and I'm not sure I even have a correct acceleration equation. But I'd like to be able to create d(t) and v(t) equations as well.

So... help? Either through equations or redirecting me to a site, where upon you may mock my google-fu.

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Re: Projectile Motion w/ Drag... help?

Postby Interactive Civilian » Tue May 19, 2009 2:00 pm UTC

Does the Wikipedia article about Trajectories help at all? It has a section on the trajectory of a projectile with air resistance.

http://en.wikipedia.org/wiki/Ballistic_trajectory
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Re: Projectile Motion w/ Drag... help?

Postby ThinkerEmeritus » Tue May 19, 2009 4:42 pm UTC

FiveFinger wrote:(Just a bit of stage setting- I am a high school student, but this is a personal inquiry and the physics teacher is out for a week...)


Your problem is more difficult than it looks at first glance. The problem is having F=-av2 , so that the problem is no longer linear in v, in which case all hell breaks loose. You can tackle it numerically if you know a programming language or can use a spreadsheet. Take a large number of small time intervals and assume that the acceleration is close-enough-to-constant over the time interval that you can use the constant-acceleration formulas. Calculate the velocity at the end of the time interval and the distance travelled over the time interval. Use the new velocity and resulting acceleration to calculate what happens in the next interval, and sum over many intervals. To check that the interval was small enough, halve it and start over (with twice as many intervals of course so that the total time is the same). If your answer doesn't change enough to bother you, the interval was small enough.

1. There are faster computational methods, but this one is easier to understand and good enough for many purposes.

2. Make sure your drag force points opposite the velocity. At first glance I am not sure that the one you listed has the right sign.

3. If you know calculus and are willing to use the less accurate F = - av, tell us, and I or someone else can quote you a form of d(t) with arbitrary constants that you can determine by differentiating twice. Knowing what form to guess for the solution is the simplest way of solving a differential equation, which is what the force law really is an example of. Guessing doesn't work so well, though, when the equation isn't linear.
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Re: Projectile Motion w/ Drag... help?

Postby danpilon54 » Tue May 19, 2009 6:25 pm UTC

I believe this problem is unsolvable exactly. We did this problem in intermediate mechanics in college and were forced to use perturbation theory to find an approximate solution, the limit of which to infinite order is the exact solution.
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Re: Projectile Motion w/ Drag... help?

Postby Shadowfish » Tue May 19, 2009 7:28 pm UTC

Interactive Civilian wrote:Does the Wikipedia article about Trajectories help at all? It has a section on the trajectory of a projectile with air resistance.

http://en.wikipedia.org/wiki/Ballistic_trajectory


This article is terrible. There is almost nothing correct in the section about motion with air resistance.
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Re: Projectile Motion w/ Drag... help?

Postby FiveFinger » Wed May 20, 2009 2:59 am UTC

Derivatives and a bit of integrals... That's as far as my calculus knowledge goes.
I know enough calculus to read almost all the equations I see, but not enough to derive them myself.

From what I'm reading though, the most accurate method is actually to use small segments of time. I will be making a simulation program to produce outputs, so that's not too much of a problem.

How much less accurate is -av as compared to -aV2? >0.1? >0.2?

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Re: Projectile Motion w/ Drag... help?

Postby Shadowfish » Wed May 20, 2009 3:24 am UTC

You definitely want to use -av^2. I don't know how much less accurate -av is in numerical way, but it should not be much harder to implement the correct formula.
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Re: Projectile Motion w/ Drag... help?

Postby FiveFinger » Wed May 20, 2009 4:09 am UTC

I'm considering using -av for the benefit of making an expression for d(t) and v(t).

However, seeing as segments of time is more accurate, I'll probably use that... good practice and all that.

It really isn't possible to make an accurate expression of d(t) and v(t)? Huh.

At least now I can feel better about having trouble finding those.

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Re: Projectile Motion w/ Drag... help?

Postby Interactive Civilian » Wed May 20, 2009 4:18 am UTC

Shadowfish wrote:
Interactive Civilian wrote:Does the Wikipedia article about Trajectories help at all? It has a section on the trajectory of a projectile with air resistance.

http://en.wikipedia.org/wiki/Ballistic_trajectory


This article is terrible. There is almost nothing correct in the section about motion with air resistance.

Fair enough. I only posted it because I remembered seeing that section when looking up something about basic ballistic trajectories. I've never read that section.

Of course, being Wikipedia, you know you are more than welcome to correct it if it is so wrong. ;)
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Re: Projectile Motion w/ Drag... help?

Postby mochafairy » Wed May 20, 2009 6:20 am UTC

You made me pull out my math text book! :shock:

I'm only going to cover 1 dimensional problems because they are easy to solve, and I hate doing eigenvalues (although, if you can find roc, he loves doing those!).

Trajectories can be modeled with differential equations. The general form of a drag problem is
[math]\frac{dv}{dt} = a - \frac{v}{\gamma}[/math]

where a is the acceleration in distance/sec2, v is the velocity in distance/sec, and gamma is the drag coefficient in mass/sec (and t is obviously the time).

In order to obtain a unique solution, v(0) has to have some value. If the object is being dropped, then it's 0. Likewise, if you throw it up at 3m/s, then v(0) = 3.

To solve the equation, first rewrite the equation as
[math]\frac{\frac{dv}{dt}}{v-\gamma a} = - \frac{1}{\gamma}[/math]
After integrating both sides, the general solution becomes
[math]ln|v-a\gamma| = -\frac{t}{\gamma}+C[/math]
Solving for v, we obtain
[math]v = a\gamma+ce^{-\frac{t}{5}}[/math]

To solve for c, simply plug in the values for v(0). This solution graphs the magnitudes of the velocity for going up or going down. After solving for v once, you need to figure out the values of t where v is 0, other than the initial point. This becomes the initial time for the second differential equation. You can plug in these new values to the general solution, to obtain the solution for the object's return trip.

If you want to plot the displacement, just remember that displacement is just the integral of velocity. Don't forget the +C!
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Re: Projectile Motion w/ Drag... help?

Postby FiveFinger » Wed May 20, 2009 2:58 pm UTC

Sorry, I think I missed something in your integration. May I show you how I proceeded through it?

(Apologies for redundancy in the solution, showing full thought process.)
[math]\frac{dv}{dt} = a - \frac{v}{\gamma}[/math]
[math]\frac{dv}{dt} = -\frac{1}{\gamma}(v - a\gamma)[/math]
[math]\frac{dv}{v - a\gamma} = -\frac{1}{\gamma}dt[/math]
[math]ln|v - a\gamma| + A = -\frac{t}{\gamma} + B[/math]
[imath]B - A = C[/imath] Decompose B and A to become C

[math]ln|v - a\gamma| = -\frac{t}{\gamma} + C[/math]
[imath]a = b[/imath] -> [imath]e^a = e^b[/imath] so...

[math]v - a\gamma = e^{-\frac{t}{\gamma} + C}[/math]
[math]v = a\gamma + e^{-\frac{t}{\gamma}}e^C[/math]
[math]v = a\gamma + ce^{-\frac{t}{\gamma}}[/math]

Where did you get the 5 in your final expression of v?
In addition, sorry to be so demanding, but could I get some kind of refrence or proof of
[math]\frac{dv}{dt} = a - \frac{v}{\gamma}[/math]
?

EDIT: Actually, I may need a full out explanation of the formula, seeing as it makes use of sec as a definition of each term, something I am completely unfamiliar with. Do you know of any reading materials you could link me to?
Second, velocity is linear rather than quadratic there. The other forum posters have made it clear that [imath]v^2[/imath] is more accurate than [imath]v[/imath]. Could you explain that as well?
Last edited by FiveFinger on Wed May 20, 2009 8:58 pm UTC, edited 1 time in total.

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Re: Projectile Motion w/ Drag... help?

Postby ThinkerEmeritus » Wed May 20, 2009 5:07 pm UTC

Sorry I was gone so long. When I went to put together a general solution, my brain complained "You haven't asked me for that kind of information since you retired! I don't remember where I put it. I need a road map to get that deep. Why don't you ask that computer you keep sending me information from?" It took a while to get my brain reoriented, and now I find that FiveFinger knows so much math that my technique may not be the best thing to suggest.

So let me comment on the accuracy of the expressions for the drag force first. I did find an article online which described an actual measurement of the drag force on a small particle. It was rather technical, but it showed a graph that looked pretty accurately F = c v2 to me. The curve was flat enough that F = bv would have been a decent approximation for smaller values of v. Usually F = c v2 is justified in textbooks by a rather rough argument, and drag is complicated enough that you wouldn't expect a single, simple formula to work in all cases. F = bv is mostly justified as a low-v approximation, and it is used solely because you can write explicit solutions for it using only elementary functions like the exponential.

Now, what I was going to do. A general solution, with 4 undetermined constants, for the problem when F = mg - b dy/dt, and using g>0 and downward positive, is y = B1 e-At + B2 + B3. Since

Ftotal = ma = m d2/dy2 and Ftotal = mg - b dy/dt

y must satisfy the equation

m d2y/dt2 + b dy/dt = mg

Differential equation theory tells us that y must have the form

y = B1 e-At + B2t + B3

and FiveFingers deserves more explanation than that, which I can do later if desired. Happily there are 4 conditions available to determine the 4 constants:

1. y(0) = y0 = the starting point
2. v(0) = dy(0)/dt = v0 = the starting velocity
3,4. Substitute into the differential equation. The resulting equation must be true at all times or we haven't gained anything. There are two functions of time in the equation, constant#1 e-At and constant#2 t. Their sum is 0, and that can be true for all time only if constant#1 and constant#2 are both equal to 0 [for instance, try t=0 and t=infinity].

For maximum satisfaction, I would suggest evaluating this form and checking numerically that it works, then using the more general form to do the numerical solution again. You might not care about the second numerical solution after doing the first one, since you will then know that you can do it if you ever need to.

Edit: I get
Spoiler:
A = b/m
B1 = -mv0/b + m2 g / b2
B2 mg/b
B3 = y0 - [-mv0/b + m2 g / b2 ]

but beware of my flaky algebra abilities. The complicated form of B3 shows that the original form could have been chosen more simply, but the choice wouldn't be obvious until the calculation was done. I hate ex post facto choices of constants.
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FiveFinger
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Re: Projectile Motion w/ Drag... help?

Postby FiveFinger » Thu May 21, 2009 12:46 am UTC

As much practice I'm getting in Calculus, I think I'm starting to lose track of the original question:

What is the best (re: most accurate) method to model projectile motion with drag, given that I have knowledge in Calculus and computation.

My apologies for letting the thread get tangential.

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Re: Projectile Motion w/ Drag... help?

Postby mochafairy » Thu May 21, 2009 12:50 am UTC

FiveFinger, there shouldn't be a 5 there. I was looking at the text and made a typo. My bad.

As far as proof goes, the one that is given in the book is basic manipulation of the "F=ma" equation. Since a, the acceleration, is the derivative of the velocity, it can be rewritten with a = v', and then the net force is the sum of the force due to gravity and the force due to drag. Dividing both sides by the mass turns it into an equation dealing with the acceleration.

This is obviously horribly simplified. The equations are assuming that the mass of the object doesn't change, that gravity is constant, and that v is at low speeds.

As far as the definitions go, sec would be seconds (but if you wanted, you could convert into different units).

If you would like some reading material, pick up a differential equations text book. The book my school uses (at least for the class I took) is Elementary Differential Equations and Boundary Value Problems, Ninth Edition by William E.Boyce and Richard C. DiPrima. I'd also suggest some physics text books, although I don't have any specific books to recommend, although some one else might be able to give you a better idea of books from the physics direction.
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Re: Projectile Motion w/ Drag... help?

Postby ThinkerEmeritus » Thu May 21, 2009 1:04 am UTC

FiveFinger wrote:What is the best (re: most accurate) method to model projectile motion with drag, given that I have knowledge in Calculus and computation.


I believe the best choice is a drag force of F = b v2 pointed opposite the direction of the velocity, with the equations of motion solved numerically using the best differential-equation solver you have access to. The constant b must be determined empirically for a body of a given size and shape.
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Re: Projectile Motion w/ Drag... help?

Postby FiveFinger » Thu May 21, 2009 2:17 pm UTC

Haha... oh wow. I thought that you had meant secant, not second.

Alright, well, thank you all for the help, calculus practice and references.


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