A car is travelling up one side of a hill and down the other side.The crest is a circular arc with radius of 45m. Determine the maximum speed that the car can have while moving over the crest without losing contact with the road.
Contact will be lost if V^2/R > g
At higher V, gravity will not provide enough centripetal force to keep it on the road.
The maximum V that allows road contact is
when V^2/R = g
V = sqrt(gR)
To determine the maximum speed that the car can have while moving over the crest without losing contact with the road, we need to consider the forces acting on the car at that point.
At the crest of the hill, there are two main forces acting on the car: the gravitational force and the normal force. The gravitational force pulls the car downwards, and the normal force (or the reaction force) acts perpendicular to the surface of the road, pushing the car upwards.
To maintain contact with the road, the normal force should be equal to or greater than zero. If the normal force becomes zero, it means that there is no upward force to balance out the gravitational force, and the car will lose contact with the road.
The maximum speed can be achieved when the car reaches the point of minimal normal force, which is when the normal force is just about to become zero. At that point, the gravitational force is providing the maximum downwards force on the car before it exceeds the capability of the normal force to push the car upwards.
To determine the maximum speed, we can use the concept of centripetal force. At the crest of the hill, the car is traveling in a circular path with a radius of 45m. The centripetal force required to keep the car moving in a circle is provided by the normal force.
The centripetal force can be calculated using the formula:
Fc = (mv^2) / r
where Fc is the centripetal force, m is the mass of the car, v is the velocity of the car, and r is the radius of the circular arc.
Since we are looking for the maximum speed, we can assume that the normal force is zero and use the gravitational force as the centripetal force. The gravitational force can be calculated using the formula:
Fg = mg
where Fg is the gravitational force and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Setting the gravitational force equal to the centripetal force, we can solve for the maximum speed (v):
mg = (mv^2) / r
Simplifying the equation, we get:
g = v^2 / r
Substituting the values, we have:
9.8 m/s^2 = v^2 / 45 m
To find v, we can rearrange the equation:
v^2 = 9.8 m/s^2 * 45 m
v^2 = 441 m^2/s^2
Finally, we take the square root of both sides to solve for v:
v = √(441 m^2/s^2)
v ≈ 21 m/s
Therefore, the maximum speed that the car can have while moving over the crest without losing contact with the road is approximately 21 m/s.