A plane loses a wheel?

Question

In testing one of the new predator drones, the engineers take the plane into a landing. At the time, the plane is at an elevation above the ground of 1100m. It takes 15 seconds for the wheel to fall to the ground.

If the plane continued as planned, it would land in exactly 5 minutes. The plane weighs 600 kg, the missing wheel and its assembly weighs 15 kg.

At what time does the plane touch ground?

((Inspired by a recent Y!A question))

Answer 1

Unless I"m missing something, the fact that the wheel takes 5 minutes to hit ground has nothing to do with this problem. Why? Because the air resistance on the wheel is different from the plane. They have different shapes. The only information that matters is the fact that the plane loses 15 kg of mass.

This is obviously not a free fall problem, neither is it a terminal velocity problem. We have to make some kind of assumption about the variable force on the wings as the plane lands. Let's say initially the plane is flying horizontally (zero vertical velocity). The net upward force F must be such that the plane begins to descend. When the plane reaches the ground we require the vertical velocity to be zero again.

So let's say F = F(v), ie the upward force depends on the velocity via some control mechanism. Then

m*dv/dt = W - F(v)

dt = m*dv / [W - F(v)]

t = m*∫{0,v}  dv / [W - F(v)] = m*g(v)

At t = 0, v = 0 and at t = T (5 mins), v = 0

So T = m*g(0)

Now without the wheel, the value of m decreases by the ratio 585/600, so we expect the new time to be

585/600 * 5 = 4.875 mins