A stuntswoman drives a bike over the top of a hill, the cross section of which can be approximated by a circle of some radius R; as shown in the diagram. She finds that, at maximum, she can drive the bike at a constant speed 120km/hr without the bike flying off the top of the hill. Use this information to find R in meters. (You may use g=9.81m\s^2)

113.26

To find the radius, R, we can use the concept of centripetal force.

At maximum speed, the stuntswoman is driving without the bike flying off the top of the hill. This means that the gravitational force acting downwards is equal to the centripetal force acting towards the center of the circle (which is the hill).

First, let's convert the speed into meters per second:
120 km/hr = (120 * 1000) / (60 * 60) m/s = 33.33 m/s (approximately)

Now, let's consider the forces acting on the bike at the top of the hill:

1. Gravitational force: The gravitational force acting downwards is given by the weight of the bike, which is equal to its mass multiplied by the acceleration due to gravity (g).
Gravitational force = m * g

2. Centripetal force: The centripetal force acting towards the center of the circle (hill) is given by the mass of the bike multiplied by its centripetal acceleration.
Centripetal force = m * a

Since the bike is moving at a constant speed, the centripetal acceleration is given by v^2/R, where v is the speed and R is the radius of the circle (hill).
Centripetal acceleration = v^2 / R

Now, equating the gravitational force and centripetal force, we have:

m * g = m * (v^2 / R)

Canceling the mass (m) on both sides:

g = v^2 / R

Substituting the known values:

9.81 m/s^2 = (33.33 m/s)^2 / R

Now, we can solve this equation for R.

R = (33.33 m/s)^2 / 9.81 m/s^2

Calculating:

R ≈ 112.3 meters (approximately)

Therefore, the radius (R) of the hill is approximately 112.3 meters.

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