ACCELERATIONS

There exist three types of accelerations:

- linear acceleration
- angular acceleration
- centripetal acceleration

Linear acceleration:

Symbol: a

Units: m/s²

Linear acceleration represents how quickly the speed of an object is changing. In physics jargon, we say it is the time derivative of the linear velocity.
That is the type of acceleration that you get when you slam on the accelerator or on the brake pedals in your car.

Angular acceleration:

Symbol: α , pronounced "alpha".

Units: rad/s²

Angular acceleration represents how quickly the angular velocity of an object is changing. In physics jargon, we say it is the time derivative of the angular velocity.
That is the type of acceleration that you get if you spin an object faster and faster, or slower and slower.
Conceptually it is more similar to linear acceleration than centripetal acceleration, because in both cases you it is about the change in how quickly an object moves: linear acceleration is about the change in how quickly an object moves along its path, angular acceleration about the change in how quickly it spins (about an axis inside the object) or rotates about some other point or axis, outside the object.

The relation between the angular acceleration and the linear acceleration is the same as the relation between the angular velocity and the linear velocity, and the same as between an angle and the corresponding arc length. As the diagram illustrate, for the same angle, the larger the radius is, the large the arc length is:

As you increase R, s increase, and it does so in the same proportion. So the relation between angle and arc length is:

s = R . θ

Linear velocity and angular velocity have the same relation (which one gets by just differentiating the equation above with respect to time):

v = R . ω

And so do linear and angular accelerations:

a = R . α

Centripetal acceleration:

Symbol: a_c

Units: m/s²

This is the one which is equal to v²/r.

Centripetal acceleration is the acceleration that represents a change in the direction in which an object is moving. It is the acceleration needed to deflect an object from moving in a straight line path.
How much an object is moving is represented by its velocity, and that has both magnitude (the speed) and direction. The change in speed is accounted for by the linear acceleration. The change in direction by the centripetal one.