Brain teaser: Lincoln Beachley

[NWilner]

Lincoln Beachley, a '20's barnstormer, raced automobiles with his airplane. The following brain teaser has fooled some
Boeing aerodynamicists:

Assume an automobile of limited power can go exactly 100 mph in still air. The car is a very sophisticated car with steel wheels and magnetic bearings, and there is zero mechanical drag, so all of the car's drag is aerodynamic. (This is, really, not a bad assumption for 100 mph anyway.) Assume an airplane of limited power can go exactly 100 mph in still air also. The airplane flys above the car and the two keep exactly even in still air. They both turn into a 50 mph wind. (Or, if you like, a 50 mph headwind suddenly materializes.) Does (a) the car gain on the airplane; (b) the airplane gain on the car; (c) they stay together?

In solving this problem, ignore any aerodynamic interaction between car and airplane, ignore any air density differences between car and airplane; there are no tricks; there are no facts unstated that are needed, there are no special geographical conditions or locations needed, there are no special effects from the earth's rotation or any special meteorological issues. You do not need to know the actual power output of each vehicle, just assume it is limited as defined.

First correct answer, SHOWING YOUR WORK, will get a prize.

 

[Jethro]

The obvious answer is they stay together. However, the fact that the car is power limited and it travels wrt the ground means that the obvious answer is not necessarily correct.

Assumptions:

The power of the car is limited.
The force output by the car is equal to the drag, which is directly proportional to the square of the velocity of the car, relative to the wind.
The new velocity of the plane will be 50 mph.

So, all we need to do is compare the new car speed to the new plane speed.

Power = F*v, so v2*(C*(v2+50)^2)=v1*C*v1^2

v1 = 100 mph, so we must solve
v2^3+100*v2^2=100^3
v2=75.5MPH

EDIT: OOPS! I goofed on my equation
It should be v2^3+100*v2^2+2500v2=100^3
v2=69.7 mph, however the conclusion still stands
So the car overtakes the plane.

 

[EvilYeti]

I don't know enough math to show my work, but I would say both vehicles would slow down, but the airplane signifigantly more than the car. This is because the airplane is using the still air for propulsion while the car is using the ground for propulsion.

Also I once saw an ultralight hover in a strong headwind .

 

[NWilner]

Extra energy?

Jethro is on top of this. As I said two Boeing aero guys missed it; and for his prize, three more questions:

1. The car's airspeed in the headwind is 75+50= 125 mph. Where does the energy come from to push the air out of the way at 125 mph while burning only "100 mph's worth" of gas?

2. What if the airplane is not an airplane but a rocket (not accelerating ambient air to generate thrust?)

3. Same result when comparing a limited power "poler" with a same speed limited power "rower", up current?

 

[Jethro]

Re: Extra energy?

Jethro is on top of this. As I said two Boeing aero guys missed it; and for his prize, three more questions:

w00t, oh, and if you happen to know of anyone looking for a recently graduated aerospace engineer...

1. The car's airspeed in the headwind is 75+50= 125 mph. Where does the energy come from to push the air out of the way at 125 mph while burning only "100 mph's worth" of gas?

Well, I suppose there's a few different ways to answer this.
The most "correct" way to say it is probably: there is no extra energy. Constant power means constant rate of energy use.
You can push the air away at 125 mph 'cause you're only pushing the earth away at 75 mph.
You're actually using MORE energy to go the same distance on the ground.

2. What if the airplane is not an airplane but a rocket (not accelerating ambient air to generate thrust?)

The thrust of a rocket depend only on the relative velocity of the exhaust gasses. Since the exhaust velocity wouldn't depend on the airspeed velocity the answer would be the same.

3. Same result when comparing a limited power "poler" with a same speed limited power "rower", up current?

Hmm, I'm not sure. I would think so, but the periodic nature of such locomotion could change things. But both are periodic, so I don't know it would change all that much. Also the drag might be linear, depending on speed, but I still don't see anything to make the answer different.

 

[NWilner]

Well, there certainly is more heat generated in pushing air at 125 than 100, yet the only power available is constant, so one needs to explain this apparent paradox.

I'm not sure what your answer was for the rocket. Same as the airplane, it loses to the car in a headwind?

Rowers are disadvantaged upcurrent for exactly the same reason airplanes are disadvantaged upwind, because the "speed" of the force applied is affected by the head wind or current. Nothing need be said about the details of the motion. It is just a matter of power and speed.

Which brings up the rocket. If it applies force independent of the airmass, why isn't it like the car? Stated another way, what is the effect of a headwind on a constant speed rocket?

 

[Skeptical Greg]

Re: Extra energy?

Jethro is on top of this. As I said two Boeing aero guys missed it; and for his prize, three more questions:

Not to detract from Jethro's contribution, but you seem to be choosing a teachers pet here..

How is this:

Jethro writes

...So the car overtakes the plane..

Different from this:

EvilYeti writes:

.... I would say both vehicles would slow down, but the airplane signifigantly more than the car.

Indeed Jethro answered first, and gets the gold star and autographed picture of Nancy Kulp, but you should give credit where credit is due.

 

[Crossbow]

OK, here is what I expect to happen:

The indicated airspeed of the airplane will be approximately 100 mph, however its speed across the ground will be reduced by approximately 50 mph.

The speed of the car would be reduced by headwind forces as well, but not as much as the airplane since the coefficent of friction is greater between the tires and ground; therefore the transmission system of the car will be better able to generate the torque needed to overcome the headwind.

Conclusion: the car will overtake the airplane.

 

[NWilner]

The emphasis was on SHOW YOUR WORK.

But the instincts were right.

On the other hand crossbow is off target because the solution has nothing to do with friction coefficients or transmission efficiencies. None of these were stated in the problem and none are necessary to solve it. In fact the assumptions were for a frictionless car with the only drag being from aerodynamic drag.

Jethro's quick solution was truly impressive.

Another question to Jethro: how did you solve the cubic? Did you solve it by algebra or brute force?

 

[Skeptical Greg]

The emphasis was on SHOW YOUR WORK.

But the instincts were right.

So, if I can do cube roots in my head, no cigar?

 

[Crossbow]

The emphasis was on SHOW YOUR WORK.

But the instincts were right.

On the other hand crossbow is off target because the solution has nothing to do with friction coefficients or transmission efficiencies. None of these were stated in the problem and none are necessary to solve it. In fact the assumptions were for a frictionless car with the only drag being from aerodynamic drag.

Jethro's quick solution was truly impressive.

Another question to Jethro: how did you solve the cubic? Did you solve it by algebra or brute force?

Hey! Just what in heck are you taking about?

How can you have a frictionless car with aerodynamic drag?

If there is drag, then there is friction.
If there is no friction, then there is no drag.

So which is it?

Also, if you are going to work out drag, and I assume that you are actually referring to parasitic drag since it is a function of speed, then you have to know a few things such as the coefficient of drag and surface area of the car and airplane you are discussing to work out the details you seek.

 

[Jethro]

Another question to Jethro: how did you solve the cubic? Did you solve it by algebra or brute force?

I pulled out my handy TI-86 and used the solve function. So brute force all the way. I am aware that a method exists for solving cubics algebraicly, but I've never actually used it.

Well, there certainly is more heat generated in pushing air at 125 than 100, yet the only power available is constant, so one needs to explain this apparent paradox.

While the rate of fuel consuption remains constant, the fuel mileage goes down.

I'm not sure what your answer was for the rocket. Same as the airplane, it loses to the car in a headwind?

Rowers are disadvantaged upcurrent for exactly the same reason airplanes are disadvantaged upwind, because the "speed" of the force applied is affected by the head wind or current. Nothing need be said about the details of the motion. It is just a matter of power and speed.

Which brings up the rocket. If it applies force independent of the airmass, why isn't it like the car? Stated another way, what is the effect of a headwind on a constant speed rocket?

The rocket is thrust limited, so it is only able to overcome a relative wind of 100 MPH.

 

[NWilner]

Jethro, I suspect you are right on the rocket but I can't prove it.

On the paradox (the car's relative wind is 125 mph, yet no more power is available), the answer is:

It is not a closed system. A PARKED car has a "relative wind" in a headwind, and energy is consumed in heating air molecules as they are deflected by the car. The "extra" energy (outside of the fuel) is wind energy.

Perhaps you'll answer crossbow's latest, since he doesn't seem to trust my answers.