Including air resistance in the calculation of falling objects

[Jdo300]

Hello all,

I have a sort of basic physics question for you all. I was wondering where I might look to get some simplified math to calculate the amounf of air resistance on a falling object if I know the shape, mass, and volume of the object. I'm doing a lab experiment where I have to drop three different sized sports balls, and my goal is to analyze the forces that are at work. Though I am not required to include air resistance, I thought it would be a fun challenge to see if I could include it as part of my data analysis. I have a basic idea of what I need to do but I'm not sure how to figure this in from the math perspective. Do I need to know information about the air environment such as air pressure and humidity?

Thanks,
Jason O

 

[dextercioby]

The first that comes through my mind is the viscous flow Stokes force (~v) which can be (not easily) computed for a spherical body moving through a Newtonian viscous fluid...And for large velocities,you might add the aerodynamical force (~v²).

If you want to find the movement as distance vs.time,you'd have to know a bit (more) of calculus...

Daniel.

 

[Jdo300]

Hmmm... I see your point. I looked up the equations you mentioned, and it looks like I'd have to understand tensors to even start to mess with that :-(. I'm just on methods of integration right now so I guess I've got a ways to go. Well, I was doing some more reading on this and allot of the resources said that a simplified way of accounting for air resistance was to just say that it increased at the square of the velocity of the falling object?? I was wondering, maybe I could somehow compute this as a related rates problem using derivatives and figure it out?? All I am basically looking for is a rough acceleration value to represent the air resistance.

 

[dextercioby]

$F_{aerodyn}=-kv^{2}$
,where "k" is a constant which depends on the transversal surface of the body and the massic volumic density of the fluid...

 

[Jdo300]

Hmm... okay. I came across this equation. Now, since my objects (except for the football) are all spherical in shape, is there something I could do mathematically to get the K value knowing information like the volume of the spheres, and the air pressure, temperature, and density of the air the balls will be fallng in?

About the transverse surface area of the body, in this case, would that just be the surface area of the spheres/ovoid? And can I compute the massic fluid densidy knowing the regular densidy of air? And if I can get these values, how do they combine to form the k constant? Also, what is the actual output of the formula? is it the force in Newtons that is exerted on the falling object?

 

[dextercioby]

k is really:
$$k=\frac{\rho}{2S}$$
Pay attention:i said CROSS SECTION AREA...Not TOTAL AREA...Assume density of air to be constant (for simplicity) and the mass volumic density of the ball is simply
$$\rho=\frac{m_{ball}}{V_{ball}}$$

Daniel.

P.S.The units for forces are Newtons...I only work in SI-mKgs.

 

[Jdo300]

Ok, thank you for your help. Two quick questions. Forst about the cross sectional area. For the footbal, would I consider the circular cross section as the area or the oval cross section as the area? My second question is are there any specific units of measure I need to use for the mass, volume and area variables? I know I'm working in metric units but would I use Liters, or mL for volume and ect.?

Thanks,
Jason O

 

[dextercioby]

Actually there needs a correction.It's the area that "feels" the air flow pointing at it.So for a sphere it's not the cross sectional area,but the area "facing" the flow.The same for every nonflat object,including an ellipsoid.

Daniel.

 

[Jdo300]

Ok... area facing the flow... so for a sphere, would this be sort of like the surface area of a hemisphere since roughly half of the sphere will be experiencing the force of the drag?

 

[dextercioby]

Exactly.Again sorry,for misleading you previously.That case would be valid for flat objects (portions of a plane)...

Daniel.

 

[Jdo300]

No problem. thank you for your help. Ok, one last question then I'll call it quits. does there have to be specific units used for the variables for the volume, mass, and surface area? I know I will be using metric units but which should I use (ex: the the volume, Liters or mL?) Or does it matter?

 

[dextercioby]

Only if you keep the line.You need't use different units.Just use the ones you were acustomed to and do it properly.

Daniel.

 

[Jdo300]

Ok, just thought of another question. I want to take that drag force in Newtons and convert it to an acceleration value. Is it valid to use the F = ma equation to solve for acceleration knowing the mass of the falling object?

 

[dextercioby]

No problem,if the mass of the falling object is constant...

Daniel.