The question can be answered both yes and no, depending on what the persons view on the english of the question is, as well as what “speed” is determined by.
The question: “If a treadmill is going the same speed in the opposite direction as a plane, will the plane take off?”
Speed on a treadmill is measured by how fast an object is going compared to the surface of the treadmill. NOT on the air speed of the object IE: A human is running 8 MPH on a treadmill set to 8 MPH. The human is sitting still in relation to the air around him. We still say “I’m running 8 MPH on my treadmill”
So, if you use that… a treadmill moving 25 MPH will have the wheels of a plane spinning at 25 MPH on top of it. The plane will be moving at 25 MPH in relation to the surface of the treadmill, but will appear to be stationary. This plane will not take off. The answer to this interpretation of the question is actually NO.
Then, people started to interpret the question to be “CAN” a plane take off, not “WILL” the plane take off. Of course a plane can take off from a treadmill. The wheels just spin on the treadmill, and the engine moves the plane forward. But, the question didn’t ask that. And by it taking off, it is moving forward, going faster than the treadmill.
The question is “EQUAL SPEED” in both directions. These people are now comparing the speed of the ground (The treadmill) to the AIR SPEED of the plane. This is not how you determine speed on a treadmill! If the plane is moving forward and the wheels are spinning on the treadmill surface, they are not equal.
IE: treadmill moving 25 MPH. plane moving 25 MPH in relation to the air. WHEELS: 50 MPH.
That just broke the rules of the question. They are not equal.
I wish people would understand that everyone interprets the question differently. It can be answered both ways, depending on the persons view.