From a college math and science teacher (I wish I could take credit, but it's not me):
"Let R be the radius of the wheels on the plane. At time t = 0, the plane applies power and proceeds to take off. Let s(t) be the horizontal component of the velocity of the plane at time t. Then the horizontal component of the velocity of the treadmill at time t is, by assumption, -s(t). The velocity of the plane relative to the treadmill is the difference of these, namely 2s(t). Now had the plane been taking off from a normal runway, the rotation rate of the wheels would be s(t)/(2πR) rad/sec; but on the treadmill the wheels will rotate at 2s(t)/(2πR) rad/sec.
At this point I shall make the assumption that the aircraft designers have designed the wheel bearings with a safety factor of > 2. The plane will now take off before the bearings burn out. Have an nice flight!"
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From a college math and
From a college math and science teacher (I wish I could take credit, but it's not me):
"Let R be the radius of the wheels on the plane. At time t = 0, the plane applies power and proceeds to take off. Let s(t) be the horizontal component of the velocity of the plane at time t. Then the horizontal component of the velocity of the treadmill at time t is, by assumption, -s(t). The velocity of the plane relative to the treadmill is the difference of these, namely 2s(t). Now had the plane been taking off from a normal runway, the rotation rate of the wheels would be s(t)/(2πR) rad/sec; but on the treadmill the wheels will rotate at 2s(t)/(2πR) rad/sec.
At this point I shall make the assumption that the aircraft designers have designed the wheel bearings with a safety factor of > 2. The plane will now take off before the bearings burn out. Have an nice flight!"