Originally Posted by MadMonkey
85 VF500F, 01 750 Ace Deluxe
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- gus gus and gidget -- birds - stewie & Libby
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Frictionless rollers!!!11one!!
(Sorry, SCCAForums inside joke. Cole fans will appreciate it)
Warning! Math and stuff ahead!
The plane will take off.
An aircraft’s engines generate forward thrust regardless of which way the wheels are turning. Let’s look at a hypothetical worst case example of this myth:
You have an endless conveyor belt moving backwards (relative to some stationary observer) at, say, 150 miles an hour. The plane is stationary relative to the belt, which means it is also moving backwards at 150 miles an hour (we’ll ignore for a moment the aerodynamic impact of trying to drive a plane backwards on a conveyor belt at 150 miles an hour). The plane needs to move forward at 100mph in order to take off. Not only can I tell you it will take off, I can tell you how long it will take.
At the beginning of our experiment, the plane has zero acceleration in either direction and a constant negative velocity of 150mph, relative to the direction we want to travel. With acceleration equal to zero the negative velocity will remain constant, and the plane will continue to remain stationary relative to the conveyor, moving backwards at 150mph.
The pilot fires up the engines, and they begin producing forward thrust. The wheels are unlocked and able to freewheel on the conveyor, and this thrust equals a force on the airplane acting in the positive direction. The airplane has mass, therefore a force acting on it produces an acceleration in the same direction (F=MA, or force equals mass * acceleration. Therefore, if you have a positive force and a fixed mass, you must have a corresponding positive acceleration. Don’t argue with me, argue with Newtonian physics). So now we have a negative initial velocity and a positive acceleration. Let’s say the acceleration is equal to 1G , or 32 feet per second per second. No, that’s not a typo, acceleration is measured in distance per unit time per unit time. Velocity is measured in distance per unit time (miles per hour, for example). So, acceleration is the change in velocity per unit time, or the rate of change of velocity.
What’s all that mean? Well, let’s do the math. Our airplane was initially moving at 150mph in the negative direction, which is 220.04 feet per second. It is accelerating in the positive direction at 32 feet per second per second. Newtionian physics gives us the following equation of motion: V=Vo + a(T-To), where V is your final velocity (our takeoff velocity of 100 mph or 146. 69 fps), Vo is your initial velocity (negative 150mph or 220.04 fps), a is our acceleration (32 fps per second), T is how much time it takes, and To is our start time (which is zero). We know everything except T, which is how long it will take to accelerate from -150mph to 100mph. So, we plug our numbers in, solve the equation, and find that T=11.46, so it will take 11.46 seconds to change the plane’s velocity from -150mph to 100mph, relative to the stationary air mass around the conveyor belt.
That answer your question?
Notice that except for giving us our initial velocity (Vo=-150mph), the velocity of the conveyor belt contributes nothing to the equation or the solution? That's because the airplane's engines are working against the air mass around the plane, which we've assumed is stationary relative to the outside observer (i.e., all velocities, both of the plane and the conveyor belt, are measured against some stationary air mass). Since the airplane's wheels are free to move at whatever speed regardless of the thrust the engines are producing, the speed of the conveyor belt has zero impact on the aircraft's ability to generate a forward thrust relative to the air around it. Once the aircraft starts to generate thrust, you have an unbalanced force on the aircraft, which results in an acceleration in the direction of the applied force. Simple physics. Then all you have to do is wait until that acceleration changes the velocity to the velocity you want, which in this case takes a bit less than twelve seconds.
