Can the Drag Equation Be Revised to Account for Velocity Differences?

[Jacob91]

I have two values for the initial velocity of a projectile through water. One is measured experimentally and the other a theoretical value for initial velocity.
The measured value is 23.75m/s
The theoretical value is 26.3m/s Resistance forces were not accounted for in this calculated value.

To account for resistance forces I can use the difference in these values.
Is it acceptable to revise the Drag Equation (Drag force=C_d* A * .5 * r * V²) to calculate the Coefficient of Drag as a function of velocity rather than force?
C_dvel = Drag velocity / (A * .5 * r * V²)
Where V=Theoretical Initial Velocity

In this way I can account for the difference in theoretical and measured initial velocity by assuming it is as a result of resistance forces. The new equation will read:
Drag velocity = C_dvel*A * .5 * r * V²

It seems like I'm probably fiddling with something I shouldn't mess with, but it does make my calculated initial velocity more accurate. Not the most elegant solution I'll admit, but it kinda works.

ANY, advice/corrections appreciated,
Thanks

 

[boneh3ad]

I think you are fiddling around with something you don't really understand. The value of the drag coefficient is that it is dimensionless and scales well across a wide range of parameters. It falls out of dimensional analysis. Your new coefficient does not share this property. I suspect the fact that it made your calculated solution more accurate is a matter of luck rather than a trend.

Other than that, you really haven't given us much information to use to help you. What is your experiment? How are you calculating your theoretical initial velocity? How are you trying to adjust it with your proposed drag correction?

 

[Jacob91]

Yep, fair point.
The project is an analysis of speargun performance.
The experiment was to measure the initial velocity of a spear as it leaves a speargun. Using slow motion video footage I have measured this velocity.

The theoretical calculation is based on the length and extension of rubber, since thick rubber bands are used to power the spear.
A tensile test of this rubber gives me the load at any extension.
I then matched this extension to the extension of the bands on the speargun.
Calculating the area under the curve gives me the energy in the bands.
From that, I've applied the kinetic energy equation (K.E. = 1/2 m v²) and using the mass of the spear and the bands(they're moving as well) I've rearranged this for velocity.

The difference in the measured and theoretical values for velocity I would like to attribute to resistance forces and apply this to the theoretical value for initial velocity for a more realistic value.

 

[boneh3ad]

Alright that makes more sense. I think you are right to consider drag to be the primary cause of your discrepancy and it sounds like you've got a pretty good handle on the stored energy in the bands. You still need to (briefly) deal in force, though.

The energy dissipated by drag is going to be the drag force integrated over the distance traveled. You'll have to account for it that way to model it correctly. You can probably make approximations to avoid the integral, such as assuming some constant average velocity (even though that's not truly correct).

The bigger problem is the drag coefficient. It's probably not something you can readily look up so you'll probably have to determine it empirically by firing the spear gun with different levels of tension and measuring the velocity. By doing that several times you can back out the drag coefficient based on the above analysis and then use it moving forward.

 

[Jacob91]

Thanks!
I'll have a play around with it. Seems like it could be quite a neat solution.
I had previously looked at using known ballistics data (G1 Reference Bullet) to calculate the coefficient of drag, but what I am modelling seemed so far removed from the experimental data I'm not sure how accurate it would have been. It also required making a number of assumptions like form factor, which was anyone's guess.

 

[boneh3ad]

That's the problem with the drag equation. There is so much physics that is wrapped up inside of $C_D$ that it's difficult at times to use. It has to be empirically determined for a good answer for a non-standard shape.

 

[RogueOne]

Yep, fair point.
The project is an analysis of speargun performance.
The experiment was to measure the initial velocity of a spear as it leaves a speargun. Using slow motion video footage I have measured this velocity.

The theoretical calculation is based on the length and extension of rubber, since thick rubber bands are used to power the spear.
A tensile test of this rubber gives me the load at any extension.
I then matched this extension to the extension of the bands on the speargun.
Calculating the area under the curve gives me the energy in the bands.
From that, I've applied the kinetic energy equation (K.E. = 1/2 m v²) and using the mass of the spear and the bands(they're moving as well) I've rearranged this for velocity.

The difference in the measured and theoretical values for velocity I would like to attribute to resistance forces and apply this to the theoretical value for initial velocity for a more realistic value.

Have you drawn a vector diagram of the band as it imparts force onto the spear? You'll want to do that with respect to the spear's displacement as well.

 

[Jacob91]

I have separated the resultant force from the bands into the X and Y components to account for losses due to the bands not being aligned with the axial direction of the spear... If that's what you mean.

 

[RogueOne]

Yep, that's exactly what I mean. Have you done it with respect to the spear's displacement as well?

 

[Jacob91]

Ah, I see.
No, I haven't.
The spear's displacement is all in the X direction since it sits on a rail mounted on top of the barrel. Any component of force in the downward Y direction is negated by this.
This would however cause frictional drag between the rail and the spear. Which I hadn't accounted for.

 

[RogueOne]

Yeah the rate of contraction of the bands will decrease, and the force will also gain bias toward the Y direction, as X approaches zero.

May the force be with you,
RogueOne