How is centripetal acceleration derived from radius and speed?

Centripetal acceleration is derived from radius and speed using the formula a = v²/r.

Centripetal acceleration is a type of acceleration experienced by an object moving in a circular path. It is always directed towards the centre of the circle and is caused by the change in direction of the object's velocity. The term 'centripetal' comes from the Latin words 'centrum' meaning 'centre' and 'petere' meaning 'to seek', which aptly describes the direction of this acceleration.

The formula for centripetal acceleration is a = v²/r, where 'a' is the centripetal acceleration, 'v' is the speed of the object, and 'r' is the radius of the circular path. This formula is derived from the relationship between the object's speed, the radius of the circle, and the time it takes for the object to complete one full revolution around the circle.

To understand this, consider an object moving in a circular path with a constant speed. Even though the speed is constant, the velocity of the object is continuously changing because its direction is continuously changing. This change in velocity results in an acceleration, which is directed towards the centre of the circle.

The magnitude of this acceleration can be found by considering the change in velocity as the object moves a small distance along the circle. If the object moves from one point to another on the circle, the change in velocity is the difference between the initial and final velocities. This change in velocity divided by the time it takes for the object to move between the two points gives the acceleration.

However, because the object is moving in a circle, the change in velocity can also be related to the change in the object's position on the circle, which is determined by the radius of the circle and the angle the object moves through. By combining these relationships, we can derive the formula for centripetal acceleration as a = v²/r.

This formula tells us that the centripetal acceleration is directly proportional to the square of the speed and inversely proportional to the radius. This means that if the speed of the object doubles, the centripetal acceleration quadruples, and if the radius doubles, the centripetal acceleration halves.