How does air resistance change with acceleration?

In summary, Mark is trying to figure out how to calculate air drag on an object undergoing acceleration, but is having trouble because the equation is dependent on velocity. He suggests using either Euler's Method or a Google search for "air resistance as a function of acceleration."
  • #1
-JammyDodger-
8
0
Hello and thanks to all who read this. Recently I've just been messing around with air drag equations, trying to extend applied maths problems to include air drag. And I've hit a road block, at least with regards to my knowledge anyway.

I've been using the F_drag = 1/2 P (mass density of fluid) v^2 C_d (drag coefficient) A (area). But my problem with this equation is it's dependence on velocity. If I were to use this equation on a projectile which is under acceleration (under gravity, and also the air drag would slow down the velocity) it would change the initial velocity, making the equation useless to me (I think).

I guessed that air drag on an accelerating body would require a differential equation, so I tried to go about making one.

F = c.v^2 (c is just the constant of pressure, area and drag coefficient etc in the drag equation)

So, I got...
dP/dt = c.v^2
m(dv/dt) = c.v^2
dv/v^2 = c/m dt

Then I went about integrating this trying to get some kind of an equation. But to no avail. I don't have a great physics knowledge as I'm only in school; so could someone be so kind as to help me get an equation which could calculate the air drag on a body that is undergoing acceleration.

I don't know if I'm making much sense in this post; but thanks anyway!

Just thinking about it there: would the best method be to calculate the air drag on the projectile at various time intervals? I.e. every second, then recalculate the air drag at the new lower speed, then, a second later recalculate again? If you get what I mean.
 
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  • #2
I'm pretty sure you can't integrate the force analytically when you have a ~v2 force term. So this would require a numerical solution to calculate the position vs. time.

Just thinking about it there: would the best method be to calculate the air drag on the projectile at various time intervals? I.e. every second, then recalculate the air drag at the new lower speed, then, a second later recalculate again?

Yes, that is the basic idea. There are algorithms of varying sophistication for doing this.

The simplest way is known as Euler's Method. Given x1 and v1, calculate x2 and v2 at a time Δt later as follows:

x2 = x1 + v1Δt + (1/2) a Δt2
and
v2 = v1 + a Δt

where "a" is calculated from F/m, given F(t, x1, v1)

Euler's Method is fairly easily entered into Excel (if you're familiar with it). A more involved, but also more accurate, method is 4th Order Runge-Kutta.

Regards,

Mark
 
  • #3
Note that your equation for drag is an approximation at low sub-sonic speeds. In the case of ballistics (bullets, cannon shells, high speed aircraft), there are no simple equations and instead tables (of coefficients) are used.
 
  • #5
Redbelly98 said:
I'm pretty sure you can't integrate the force analytically when you have a ~v2 force term. So this would require a numerical solution to calculate the position vs. time.



Yes, that is the basic idea. There are algorithms of varying sophistication for doing this.

The simplest way is known as Euler's Method. Given x1 and v1, calculate x2 and v2 at a time Δt later as follows:

x2 = x1 + v1Δt + (1/2) a Δt2
and
v2 = v1 + a Δt

where "a" is calculated from F/m, given F(t, x1, v1)

Euler's Method is fairly easily entered into Excel (if you're familiar with it). A more involved, but also more accurate, method is 4th Order Runge-Kutta.

Regards,

Mark

Dont do this. It won't work
 
  • #6
One site with interesting explanations and formulas is

http://www.math.cornell.edu/~numb3rs/lipa/end_of_watch.html

You want the formula under VERTICAL MOTION which reflects acceleration due to gravity... but this formula is for uniform (constant) acceleration...I assume that's what you want...you'll note there the drag force is an exponential function of time and drag coefficient.

OR ...Try Googling "air resistance as a function of acceleration" ..lots of hits...
 

What is air drag?

Air drag, also known as air resistance, is a force that acts on an object as it moves through the air. It is caused by the friction between the object's surface and the air molecules.

How does air drag affect acceleration?

Air drag can decrease the acceleration of an object by opposing its motion. As the object moves faster, the air resistance increases, which leads to a decrease in acceleration.

What factors affect air drag?

The factors that affect air drag include the shape and size of the object, the density and viscosity of the air, and the speed and direction of the object's motion.

How do you calculate air drag?

The air drag force can be calculated using the formula F = 0.5 * ρ * v^2 * Cd * A, where ρ is the density of air, v is the velocity of the object, Cd is the drag coefficient, and A is the object's frontal area.

How can air drag be reduced?

Air drag can be reduced by changing the shape of the object to be more streamlined, reducing its frontal area, or by using materials that minimize the friction between the object and the air.

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