View Single Post
Old 03-03-2006, 05:11 PM
timmerov2 timmerov2 is offline
Join Date: Mar 2006
Posts: 2
Originally Posted by Saltire
It sort of frightens me to even post on one of these threads, but I'm going to try one more way to get people to BR#1.
heh. you said it.
there's another very big difference between an airplane and a car with wings. the speedometer in a car is attached to the wheels. the "speedometer" in an airplane is called an airspeed indicator and is attached to tubes that stick out into the airstream and measure the speed the air is moving over the wings. oddly enough.
there are many speeds in this problem. let's write them all down.
va = airspeed = speed of the airplane through the air.
vg = ground speed = speed of the airplane relative the ground.
vs = "speedometer" speed - assume our airplane is equipped with a speedometer = tire speed.
vb = belt speed = speed of the treadmill's belt. the top of the belt travels at vb and the bottom of the belt travels at -vb.
and how do these speeds relate?
let's assume for simplicity that there's no wind.
va = vg
the speedometer speed will be the air/ground speed plus the belt speed.
vs = va + vb
now... the problem as stated doesn't specify which speed the treadmill is using to set its belt speed.
group #1 assumes they mean the airspeed, va (=vg). which is really the only meaningful speed for an airplane. and they conclude that the airplane pretty much ignores the treadmill and takes off.
group #2 assumes they mean speedometer speed, vs. and there's no difficulty with that so far. the problem then introduces a constraint: the belt speed is set to the speedometer speed. in equation form:
vb = vs
but from above we know that:
vs = va + vb
va = 0
and the airplane doesn't take off. it's simple math, right?
well... the math is correct. but the conclusion isn't.
the constraint can be restated like this: what has to happen in order for the belt speed to be equal to the ground speed?
the conclusion is the same:
va = 0
ie the airspeed is zero. but now the interpretation is different. this time around it's clear we need to tie the airplane to the ground to set va=0. and the airplane doesn't take off. oddly enough. note that once you set va=0, you can run the belt at any speed you want. it will always match the plane's speedometer. however, if you let the plane move, ie va!=0, then you cannot set the belt speed to the speedometer speed no matter how fast you turn the belt.
group #2 runs into trouble when they draw their conclusions. the wrong conclusion is that the treadmill somehow keeps the plane from taking off. the right conclusion is that the only way to set the belt speed according to the stated problem is to prevent the plane from moving. note that in this case it doesn't matter that it's an airplane. a car, cat, person, author would be equally restrained.