Schoolyard Science #3. Take Off!

  

Okay class, We're revisiting  altitude and atmosphere...

Quick review:
Bogota is one of the highest cities in the world  (Denver's at 5,300 ft.  Bogota's at 8,300 ft / 2,500 meters.) Check out the graph from Wikipedia...
Bogota has about 77% of the air that Miami has.

[Special Shout Out to Buck Buchanan, ace instructor at Airbus.  Thanks, Buck!
Any errors in this blogpost are mine. Buck did not check my final posting.]

We've been on a couple of airplane flights between Bogota and cities along the coast, and we made some interesting observations.
First a little background music...
When a jet plane accelerates to take off, it grabs air from in front of the engines, and shoves it out the exhaust at very high speeds.  (There's other stuff going on too, but that's the big idea.)  The accelerated air causes the plane to move forward. (No.  The exhausting air doesn't "push" on the static outside air.  See me after class if you have questions about this.)
The plane builds up speed, and when it reaches 150 knots, the pilot tips the plane up and the plane takes off.  The pilots call this speed "VR".  At sea level, this acceleration takes about 30 seconds to get from zero speed to 150 knots (VR).  Here a sketch from Wikipedia.

So what's this got to do with Bogota?

Here's a photo of a timer I used to record how long it took to get airborne from an airport near sea level.

That's cool. About 33 seconds to take off at sea level. 

Yawn.

Ok Class. But what happens if the air is less dense, say if you landed in Bogota and your now going to take off?
- Does it take longer to accelerate to 150 knots? Less time?
- Do you accelerate faster because there's less wind resistance (the air is thinner.)?
- Do you need a longer runway or a shorter runway?  (Said another way...Will you run into the nursery school that's at the end of the runway? Ugh...)
- Is 150 knots the speed you need? More or less?

Here's the answer, from a recent take off from Bogota's airport.

Whoa!
Taking off in Bogota takes about 40% more time than in Miami! (46sec/33sec = 140%).
It's true, because the planes in Bogota accelerate more slowly than they do in Miami...even when the engines are running at the same speed.  (We checked :-)  Two engine aircraft always set their throttles to the same value for take-off*.  That way the engines are most efficient and operate most cost-effectively.)

But wait.  Does more time needed to take off mean planes require more distance to take off?  After all, they're moving more slowly, right?   The pilots of the plane told me that take off speed was about the same for Miami and Bogota (150 knots). (Yes.  I asked them. And you can check it out here.)

For any former high school physics students, here your challenge question:
Assume that:
1. The take off speed for an airplane is the same at any altitude.
2. And at a higher runway altitude it takes 46 seconds to reach take off speed, and at sea level it takes 30 seconds to reach that speed.
3. Acceleration from break release through take off is constant.

Tell us:
- How long is the takeoff distance in Bogota compared to Miami? Show your work.
Post your answers on this blog. :-)  It's okay to show off.
(Hint: Use a graph of velocity vs. time.)

Bonus question.
Now, Why does the plane accelerate more slowly?  Show your thinking. :-)
Hint: F=ma.
First think about the mass of the air that's going into the engine.
Assume that the air accelerates from zero at the engine intake to some exhaust velocity. (It doesn't matter what the actual velocity is.)
Assume that the engine's rotation speed is the same in both scenarios.  (It is.)
Next make a simplifying assumption about the mass of the fuel that's added to the engine.
Compare this generated force (which is now acting on the mass of the airplane) with the acceleration of the plane in the two altitude scenarios.  Go ahead, you can do it.

Your answer won't come out exactly, because there are other factors that affect the take off time, buot I think you'll be impressed at how close you'll get.  Good luck!