A motorcycle is traveling up one side of a hill and down the other side. The crest of the hill is a circular arc with a radius of 43.0 m. Determine the maximum speed that the cycle can have while moving over the crest without losing contact with the road.

To determine the maximum speed that the motorcycle can have while moving over the crest without losing contact with the road, we can use the concept of centripetal force.

When the motorcycle is at the crest of the hill, the gravitational force acting on it can be resolved into two components: one perpendicular to the surface of the hill, and the other tangential to the circular path. The perpendicular component provides the normal force, which is responsible for keeping the motorcycle in contact with the road.

At the maximum speed, the normal force will be zero, as the motorcycle is just about to lose contact with the road. So, we need to equate the gravitational force and the centripetal force to find the maximum speed.

The gravitational force acting on the motorcycle at the crest of the hill can be calculated using the formula:

F_gravity = m * g

where m is the mass of the motorcycle and g is the acceleration due to gravity (approximately 9.8 m/s^2).

The centripetal force can be calculated using the formula:

F_centripetal = m * v^2 / r

where v is the velocity of the motorcycle and r is the radius of the circular arc (43.0 m).

Setting the gravitational force equal to the centripetal force:

m * g = m * v^2 / r

Canceling the mass (m) on both sides:

g = v^2 / r

Rearranging the equation to solve for v:

v^2 = g * r

v = sqrt(g * r)

Now, substitute the values for g and r:

v = sqrt(9.8 m/s^2 * 43.0 m)

v ≈ sqrt(421.4) m/s

v ≈ 20.5 m/s

Therefore, the maximum speed that the motorcycle can have while moving over the crest without losing contact with the road is approximately 20.5 m/s.