How does air resistance change with acceleration?

Click For Summary

Discussion Overview

The discussion revolves around the effects of air resistance on a projectile undergoing acceleration, particularly in the context of applying air drag equations to problems in applied mathematics. Participants explore the complexities of integrating air drag into motion equations when acceleration is present.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant expresses difficulty in using the drag equation due to its dependence on velocity, suggesting that a differential equation might be necessary for an accelerating projectile.
  • Another participant agrees that analytical integration of the force with a velocity-squared term is not feasible, advocating for a numerical solution to track position over time.
  • Euler's Method is proposed as a simple numerical approach to calculate position and velocity at discrete time intervals, with a more accurate alternative suggested as the 4th Order Runge-Kutta method.
  • A participant notes that the drag equation is an approximation valid only at low sub-sonic speeds, indicating that more complex scenarios, such as ballistics, require different approaches, including the use of tables for coefficients.
  • One participant mentions that a falling object will eventually reach terminal velocity, assuming it starts from a sufficient height, referencing external material for further reading.
  • Another participant provides a link to a resource with formulas for vertical motion under constant acceleration, noting that the drag force can be modeled as an exponential function of time and drag coefficient.
  • There is a caution against using certain methods, with one participant stating that a proposed approach would not work.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best method for calculating air drag on an accelerating body. Multiple competing views and methods are presented, with some participants agreeing on the need for numerical solutions while others highlight limitations of existing models.

Contextual Notes

There are limitations regarding the assumptions made about the drag equation's applicability, the dependence on velocity, and the conditions under which different methods may or may not work. The discussion also reflects uncertainty about the effectiveness of various numerical methods in this context.

-JammyDodger-
Messages
8
Reaction score
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.
 
Last edited:
Physics news on Phys.org
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
 
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.
 
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
 
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...
 

Similar threads

Undergrad Express drag force/acceleration/velocity as function of time
  • · Replies 14 ·
Replies
14
Views
2K
Undergrad Fluid dynamics: drag coefficient and pressure at the stagnation point.
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Drag Coefficient of a 'flexible' object (e.g. a piece of paper)
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Drag equation - relative flow velocity
  • · Replies 11 ·
Replies
11
Views
4K
Undergrad If air drag is a factor, how will it affect a projectile?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Air drag force: Why heavy objects accelerate faster?
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Help with drag coefficient calculated from ballistic coefficient
  • · Replies 3 ·
Replies
3
Views
9K
High School Problems with modelling the vertical motion of a dust particle
  • · Replies 15 ·
Replies
15
Views
2K
Undergrad Relationship between Drag Force & Viscous Torque on USA Football
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad General expression of wind force on a sail....
  • · Replies 40 ·
2
Replies
40
Views
8K