# Relationship between Drag Force & Viscous Torque on USA Football

TomB
Hello,

I am a Physics undergrad degree graduate who is attempting to do a personal project involving the projectile motion of an American football in a 2D space. I want to take into account the drag force that the ball will experience and also take into account the spin of the football (in the same direction of the velocity of the ball) which should have some sort of relationship with the drag. I assume the ball is thrown with no wind and the ball is thrown through air at 20 Celsius.

Let x be the horizontal coordinate and z be the vertical coordinate.

I believe I have two equations to start:

m * a_x = C * v_x
m * a_z = C * v_z - m * g

where
m is mass of football
a_x is acceleration of football in x dimension
C is drag coefficient
v_x is velocity of football in x dimension
a_z is acceleration of football in z dimension
v_z is velocity of football in z dimension
g is acceleration due to gravity

Firstly, is it right for me to assume that C will be the same in both directions considering the non spherical shape of an American Football?

Next, I am curious if there is some way to build out the relationship between C (drag coefficient) and w (angular velocity of football in direction of velocity of football) I assume is proportionate to w in some way, but I am unsure how to get started in terms of finding this relationship.

If anyone can give me any tips, papers, textbooks or anything that can get me on the right direction for this project please let me know.

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I don't know a specific textbook, but there are some titled "The physics of sports". I'm pretty sure ther's one treating the flight of an American football, but that's for sure quite a challenging problem since it's a symmetric top (but not a spherical one as with spherical balls), and I guess you need full 3D. Treating this including air resistance and spin is for sure a great but challenging subject. I guess, at one point you should also consider using numerics.

Perhaps the physics of baseball or golf with a round ball is also a good first step to start. Also there you have air resistance and spin (including the Magnus effect), etc. Also here I guess for the most realistic situations you'll need numerical calculations for solving the equations of motion.

Arjan82
So I think I can assume that the ball's long axis (from point to point) is always in the direction of flight like I think I see when looking at some videos of this? So when the football is following a parabolic curve, the long axis is always tangential to that parabola? (Sorry, in Europe we don't do American football ;)

Some thoughts:
In the case the orientation of the long axis of the ball is not constant with respect to the axes, C differs not only between x and y direction, but also from time to time since the ball is rotating. But it definitely differs between the direction parallel to the long axis and orthogonal to the long axis.

I don't see w influencing C by that much if w is neatly around the long axis of the ball? Mainly because the velocity of the surface due to the rotation is at a right angle with the flow direction. I'm not sure though, why do you think there is an influence?

TomB
So I think I can assume that the ball's long axis (from point to point) is always in the direction of flight like I think I see when looking at some videos of this? So when the football is following a parabolic curve, the long axis is always tangential to that parabola? (Sorry, in Europe we don't do American football ;)

Some thoughts:
In the case the orientation of the long axis of the ball is not constant with respect to the axes, C differs not only between x and y direction, but also from time to time since the ball is rotating. But it definitely differs between the direction parallel to the long axis and orthogonal to the long axis.

I don't see w influencing C by that much if w is neatly around the long axis of the ball? Mainly because the velocity of the surface due to the rotation is at a right angle with the flow direction. I'm not sure though, why do you think there is an influence?

Thanks for getting back to me.

Yes you can assume that the ball's long axis is always in the direction of flight. Yes, the assumption is that the long axis is always tangential to that parabola.

Makes sense on the C. The different surface area / shape definitely would make sense as a reason for different C.

I found this random/non scientific clipart to make sure we are on the same page. Using the right hand rule I believe the direction of the angular velocity would be the same as the direction of the flow. My thought is that the spinning will contribute to lower drag effects. Perhaps considering I am assuming 0 wind and low viscosity fluid it will not make a difference in my system, but my thought was that a higher angular velocity would help "cut" through the air and the body would experience a smaller drag.

Let me know if this makes sense.

Arjan82
Actually, the only way I see w influencing C is when the flow separation at the back end of the ball is somehow influenced. If this is somehow reduced, such that the pressure recovery at the aft part of the ball is larger, then you would get a reduction in drag. However, it might be that rotation also increases the pathlength of the flow in the boundary layer of the ball, meaning frictional resistance increases. It is complex...

Also, computing this is not straight forward at all, also not with numerical methods. This is because the roughness elements on the surface of the ball (the dimples, stiches, ridges etc.) significantly influence flow separation as well, this is not easily computed with numerical methods... For an accurate result you need someone to have measured C as function of w I think.

However, if you just want to toy around with formulas and see the effect for some assumed influence of w on C, you can always simply assume a linear, quadratic or other type of relation between them.

• TomB
TomB
Actually, the only way I see w influencing C is when the flow separation at the back end of the ball is somehow influenced. If this is somehow reduced, such that the pressure recovery at the aft part of the ball is larger, then you would get a reduction in drag. However, it might be that rotation also increases the pathlength of the flow in the boundary layer of the ball, meaning frictional resistance increases. It is complex...

Also, computing this is not straight forward at all, also not with numerical methods. This is because the roughness elements on the surface of the ball (the dimples, stiches, ridges etc.) significantly influence flow separation as well, this is not easily computed with numerical methods... For an accurate result you need someone to have measured C as function of w I think.

However, if you just want to toy around with formulas and see the effect for some assumed influence of w on C, you can always simply assume a linear, quadratic or other type of relation between them.

Thanks! Yeah that makes sense.. .this may be a more complex problem than I initially thought. It is not a bad idea to attempt to experimentally determine the relationship between the two and figure out the relationship.

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