How can I show the drag force-velocity relationship of a shuttlecock from my recorded trajectories?

[qumbo19]

For a school project, I’m looking at modelling the vertical trajectory of a shuttlecock. I have several videos of trajectories, along with velocity and position data from LoggerPro. I’m aware that for a shuttlecock, the drag force is proportional to velocity squared, but is it possible for me to show this from my trajectory data?

 

[Lagrange fanboy]

The difference between the rate shuttlecock decelerates and the expected rate due to gravity will be due to drag. You might also find motion history image useful.

 

[Chestermiller]

For a school project, I’m looking at modelling the vertical trajectory of a shuttlecock. I have several videos of trajectories, along with velocity and position data from LoggerPro. I’m aware that for a shuttlecock, the drag force is proportional to velocity squared,

Sez who?

Plus, the drag behavior is going to be anisotropic, depending on the angle of the relative velocity with respect to the axis.

 

[haruspex]

For a school project, I’m looking at modelling the vertical trajectory of a shuttlecock. I have several videos of trajectories, along with velocity and position data from LoggerPro. I’m aware that for a shuttlecock, the drag force is proportional to velocity squared, but is it possible for me to show this from my trajectory data?

By "vertical trajectory", do you mean that the trajectory is purely vertical in your experiment or that you are modelling the vertical component? If the latter, you need to realise that the horizontal and vertical motions cannot be decoupled under quadratic drag. The greater the horizontal velocity, the greater the total drag force, and the greater the vertical component of the drag.

Whether the drag is approximately quadratic depends on the speed. At low speeds it is closer to linear, and in that phase the horizontal and vertical motions can be decoupled. You might be able to detect the transition in your data.

 

[qumbo19]

By "vertical trajectory", do you mean that the trajectory is purely vertical in your experiment or that you are modelling the vertical component? If the latter, you need to realise that the horizontal and vertical motions cannot be decoupled under quadratic drag. The greater the horizontal velocity, the greater the total drag force, and the greater the vertical component of the drag.

Whether the drag is approximately quadratic depends on the speed. At low speeds it is closer to linear, and in that phase the horizontal and vertical motions can be decoupled. You might be able to detect the transition in your data.

I mean the vertical component, not just dropping a shuttlecock vertically. Is it not possible to get some reasonable approximation for the vertical component of the trajectory? I managed to get the velocity time by integrating -mg-bv^2 = ma. Is this not a valid approach?

 

[haruspex]

I mean the vertical component, not just dropping a shuttlecock vertically. Is it not possible to get some reasonable approximation for the vertical component of the trajectory? I managed to get the velocity time by integrating -mg-bv^2 = ma. Is this not a valid approach?

No.
Suppose at some point the trajectory is down, at $\theta$ to the vertical, speed v.
The drag force is $bv^2$, upward at $\theta$ to the vertical.
The vertical component of that is $bv^2\cos(\theta)$, leading to $m\frac{d(v\cos(\theta))}{dt}=mg-bv^2\cos(\theta)$.
Or, in terms of the velocity components, $m\frac{d(v_y)}{dt}=mg-bv_y\sqrt{v_x^2+v_y^2}$.
Note, e.g., that if the horizontal component is large enough then the downward velocity diminishes instead of increasing.

 

[vanhees71]

I mean the vertical component, not just dropping a shuttlecock vertically. Is it not possible to get some reasonable approximation for the vertical component of the trajectory? I managed to get the velocity time by integrating -mg-bv^2 = ma. Is this not a valid approach?

Hm, for a full solution I'd rather numerically integrate the equations of motion. I guess a realistic description of a shuttlecock is pretty difficult. It's easier for a baseball or golf ball. There you can have all kinds of sophistication like taking into account spind and the Magnus effect :-).