Predicting how far an object will fly

[askphysicsquestions1]

TL;DR

I am using an air nozzle to blow pressurized air onto a metal piece to blow it off of a table, about 100mm off of the ledge. I figured it would be pretty simple to model how far it will blow off of the table, but having a lot of trouble

Hi,

I have an experimental setup where we are taking certain different types of metals of varying shapes and sizes, weighing them, taking approximate measurements, and then blowing it off of a table of a fixed height with an Air Nozzle. The data taken down in experiment is the PSI at which the air nozzle is set to; how far the piece flew (horizontally from base of the table), weight of the piece, and of course the table height is fixed.

I wanted to try and compare that to predictions via formula; factoring in air resistance:

The vertical component of the movement was modeled by Net Fy: mg - k(Vy)^2

Fy = Force in y direction
mg = mass X gravity (or simply just the weight in kilograms)
k = a constant; that includes the coefficient of drag
Vy = Velocity in y direction

Solving that differential equation gave me:

Vy = sqrt(mg/k) * tanh(sqrt(gk/m) * t)

Integrating:

Xy = sqrt(mg/k) * ln cosh(sqrt(mg/k) * t) + ln(2) ... this is the vertical distance. Xy would be the height of the table and would solve for t

Acceleration off of the table:

Fx = ma = Fnoz - u*mg; solve for a

Fx = Force in x direction (ma is the net force)
Fnoz = Force of the nozzle. This was calcualted by taking pi*r squared times the air pressure in bar. The data sheet also recommended this method
u = coefficient of Kinetic Friction
a = horizontal acceleration

Predicted Distance:

X = sqrt(ma/k) * ln cosh(sqrt(ma/k)*t) + ln(2)

Using this method, I predicted distance...

And was way way way off...

We took about 600 pieces of data of various different metals. I don't have any data for k or u... i intended on adjusting k & u for each metal until the answer on a few rows is close to correct. I assumed once I was in the neighborhood, then other rows for that metal would start getting close too... especially since I think k isn't supposed to change much for the same metal on the same surface, and through air.

Possible Errors:

- Conceptual errors on my part, mathematical errors that I haven't caught.

- Acceleration off of the table; the piece gets blown on by the nozzle about a 100mm off of the ledge of the table. The nozzle gets pressed electrically by a human pressing a button. I would imagine that the amount of time it takes to press a button and release stopping the air flow, the piece has alraedy traveled off the table, so the acceleration of the net force from the nozzle minus the friction would be still valid.

- Force of the Nozzle. Let's say we set the compressor to 4 PSI; then we took the force to be 5.027 N (the nozzle opening had a radius was 4mm, so took =4 PSI*100000*(PI()*(0.002)^2) ). That seems like too much force

I can go into more about calculations and values and units in the comments, but figured I would start with this in case someone can point out some conceptual and obvious errors I may have missed.

 

[jbriggs444]

If I understand correctly, you are assuming that horizontal nozzle air flow pressure continues to act on the object throughout its fall, resulting in a constantly accelerating forward horizontal speed. Further, in spite of the forward horizontal air flow, it seems that the falling particle is still (per the given formula) subject to backward horizontal wind resistance. Surely that cannot be the intent!?

You should be aware that with quadratic drag, the horizontal and vertical differential equations are not independent. They are coupled. Vertical air resistance is affected by horizontal speed. Horizontal air resistance is affected by vertical speed.

We have been given essentially zero information about the experimental setup. In my mind's eye, we would have a flat surface of the table over which the object is first blown before reaching the lip. After the lip, the object would drop out of the flow stream and would continue in a ballistic trajectory (albeit subject to wind resistance).

Since nozzle to table lip distance is not listed as a parameter in your equations, perhaps the actual setup is otherwise. Please enlighten us.