Physics: Deceleration to Terminal Velocity

I hope this makes sense…

My colleagues and I were discussing the speed of a grain of wheat (or rye) being discharged from an airplane flying 150 miles per hour. There is actually quite a lot of research on the terminal velocity of grain. Wheat is approximately 7.5 to 8.12 meters per second which is about 17 miles per hour.

Everyone could agree the grain would drop at 7.5 m/s, but we couldn’t come to an agreement on how fast the grain would decelerate along the horizontal axis. Then there was talk of horizontal terminal velocity…which is beyond me, but I think it is the same.

Here is a real world scenario (with estimated numbers): An airplane may fly 150 mph and drop seeds over a standing crop 30 feet above the ground. A helicopter may only be able to do it at 100 or 120 mph.

Will the seed from the airplane have more horizontal penetrating power into the standing crop? Can you calculate the horizontal air drag on the grain and chart how quickly it decelerates? It would be interesting to know if the grain is traveling 85 mph when it is applied at 30 ft and how it would change if the aviator was higher or lower.

It’s basically the same as firearm ballistics (or any ballistics, I guess). A bullet will leave the muzzle at speeds much beyond it’s terminal velocity, and will drop more the further it flies as it slows.

There are no end of programs to calculate the bullet drop. They often give the drop as well as the velocity at any specified distance. I’ll try to dig one up later.

The similarity to firearm ballistics was mentioned in our discussion. I felt like the ballistics model may not account for the drag on a somewhat irregular shaped object, but I could be wrong about that.

The only reason there’s a non-zero terminal speed in the vertical direction is because the force of drag is balanced by the force of gravity in that direction at that speed. In the horizontal direction, there is no constant force as there is in the vertical direction, so the terminal horizontal speed (in the absence of wind) is zero. How quickly the object approaches zero horizontal speed depends on the aerodynamic properties, of course.

If you are comparing an airplane vs. helicopter, the downdraft from the helicopter might be a more significant difference. You would feel it if you were standing on the field below, right? You’d probably want to add that vertical wind speed to the terminal velocity. (Terminal velocity is airspeed relative to the surrounding air.)

That is a good point. The cover crop is transported in a basket that hangs 15 or 20 ft below the helicopter. I am curious as to the amount of down draft there is. I wonder if it is significant considering the down draft doesn’t affect the standing crop.

Here’s an on line ballistic calculator. One of the variables is “BALLISTIC COEFFICIENT”. Obviously, for something as variable as a biological sample, the variable is going to be very variable. You’d probably want to enter a range of values. If you want the underlying formulas, you can find them here

Something else that will confound modeling is turbulence. A single grain of wheat is one thing, but 100,000 flying in formation is something else.

Another point to be aware of: Projectile motion in a vacuum can be broken up into a horizontal component and a vertical component, with the two components solved separately, but this cannot be done when you have aerodynamic drag. Horizontal motion will increase the vertical component of drag, and vice-versa.

This will really only be an issue if the helicopter is hovering. If the heli is in forward flight with an airspeed comparable to an airplane, then it will generate a vortex pair similar to that plane. In other words, yes, there will be a downdraft, but nothing like the violent column of air you observe during a stationary hover.

I think the issue with a massive release of grain isn’t turbulence, it’s entrainment. The difference between 1 grain and 100,000 is like the difference between one snowflake and an entire mountainside of snow: in the latter case, the total mass and drag of all the individual particles get the entire air mass moving, which reduces drag on the particles, enabling a higher sustained ground speed than you would see with one single particle. If OP is releasing grain like this, I think they would be observing relatively uneven grain distribution on the ground.

EagleEye, can you provide some typical measurements for a grain of wheat, i.e. mass and length/width?

Look up the physics of the Norden bombsight. It is basically doing the same thing. Different coefficients of drag, but the same equations.

IANAPhysicist, but this occurred to me to. Modeling this in theory must be a nightmare. Turbulence regimes/Reynolds numbers galore.

Lots of good info here! Yes I would imagine there are a lot more complex equations at play here. I would say a single grain is about 2-3 mm wide by 4-5mm long and we are applying about .75 million per acre. Not to mention additional radish seed.

I also think that the way each machine discharges could be at play. The airplane discharges to the back, while the helicopter sort of flings grain everywhere. We have a very unscientific feeling the airplane is more consistent.

Thanks for the pics. This site won’t display images in posts, so I’m repasting your links here without the img tags:

Helicopter seed dispersal

Fixed-wing seed dispersal

A couple of questions:

-Is there some kind of broadcast spreader on the bottom of the helicopter bucket? It looks like the seed is getting flung out in all directions, instead of just down and to the rear.

-The fixed-wing plane looks like an ordinary crop duster that sprays liquid from nozzles on the wings. Does this one disperse seed the same way, or is the seed only dispensed from a single nozzle on/near the fuselage?

In looking at the helicopter image, it looks like the avalanche-type entrainment of air won’t be an issue, and we can think of each grain as acting independent of other grains.

I have a ballistics calculator spreadsheet, and I will fiddle with it later today or tomorrow and see what kind of behavior falls out of it for various initial forward speeds.

Thank you for properly posting the images!

The helicopter does appear to randomly fling seeds.

The airplan has a triangular set of tubes below the fuselage next to the wheels that disperse seed over 100-120 ft.

Here are a few other pictures that I think might show what you’re talking about:

Exhibit A

Exhibit B

Exhibit C