A pilot flies his plane in a vertical loop of radius 500 m at such speed that at the top of the loop he feels no force from either the seat or the seatbelt. At what speed is the plane flying?

At the top of the loop, the pilot is experiencing weightlessness, which means that the centripetal force is equal to the gravitational force acting on the pilot.

The centripetal force required to keep the pilot moving in a circular path at the top of the loop is provided by the lift force generated by the wings of the plane. The lift force is given by the equation:

Lift force = Mass x Centripetal acceleration

Since the pilot is experiencing weightlessness, the lift force is equal to the gravitational force acting on the pilot:

Lift force = Mass x g

Where:
g = acceleration due to gravity = 9.81 m/s^2

The centripetal acceleration is given by:

Centripetal acceleration = v^2 / r

Where:
v = speed of the plane
r = radius of the loop = 500 m

Setting the lift force equal to the gravitational force and substituting the expressions for lift force and centripetal acceleration, we get:

Mass x g = Mass x v^2 / r

Solving for v, we get:

v = sqrt(g x r)
v = sqrt(9.81 m/s^2 x 500 m)
v = sqrt(4905 m^2/s^2)
v = 70 m/s

Therefore, the plane is flying at a speed of 70 m/s.

The pilot feels no force at the top of the loop because the normal force exerted by the seat is equal in magnitude but opposite in direction to the gravitational force acting on the pilot. At this point, the pilot is in circular motion, so the sum of the forces acting on him must be equal to:

$$m\frac{v^2}{r}=mg$$

where:

- m is the mass of the pilot
- v is the speed of the plane
- r is the radius of the loop
- g is the acceleration due to gravity

Solving for v, we get:

$$v=\sqrt{gr}$$

Substituting the given values, we get:

$$v=\sqrt{(9.81\text{ m/s}^2)(500\text{ m})}=70.7\text{ m/s}$$

Therefore, the plane is flying at a speed of 70.7 m/s.