# A car drives over the top of a hill that has a radius of 50 m .

General guidance

Concepts and reason

The main concepts required to solve this problem are the Newton’s second law and the centripetal force.
Initially, apply the Newton’s second law on the system and then equate both the force due to gravity and the centripetal force. Finally, rearrange the expression for the maximum velocity.

Fundamentals

The force is defined as the product of the mass of the object and the acceleration.
The weight W of the object is,

Here, W is the weight of the object, m is the mass of the object, and g is the acceleration due to gravity.
The expression of the centripetal acceleration is,

Here, is the centripetal acceleration, m is the mass of the object, v is the velocity of the object, and r is the radius of the circular path.

Step-by-step

Step 1 of 2

The equation of motion is,

Here, mg is the weight of the car, N is the normal force acting on the car from the ground, m is the mass of the car, v is the velocity of the car, and r is the radius of the circular path.
If the car will fly off the ground, then the normal force N should be zero.
Substitute 0 for N to find the maximum speed allowed.

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Rearrange the equation.

The equation of the motion for the system of car on the top of a hill can be written using the Newton’s second law. The expression of the maximum velocity of the car without flying off the road at the top of the hill can be calculated using the force due to gravity and centripetal force.

The value of the normal force is zero, if the car will fly off the ground. So, do not take any non-zero value.

Step 2 of 2

The expression of the maximum velocity is,

Here, g is the acceleration due to gravity and r is the radius.
Substitute 50 m for r and for g.

The maximum speed of the car is 22 m/s

The maximum speed of the car is 22.1 m/s. The speed of the car must be less than the maximum speed so that the car will not fly off the road. If the speed exceeds this maximum value, then the car will fly off the road. The maximum speed of the car can be calculated using the acceleration due to gravity and the radius.

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The maximum speed of the car is 22 m/s