Clarification(tangential acceleration)

[negation]

As has been, accceleration is always the derivative of velocity.
In rotational motion (circular uniform), however, the tangential acceleration is the change in speed rather than velocity. Why is this so?

For any points on a uniform circular disc, the tangential velocity is changing because direction is changing but the magnitude of the tangential velocity (AKA speed) is constant. Well, make sense, the quantity remains the same but the direction changes and always in a direction tangent to the centripetal acceleration.
Why then is tangential acceleration make in reference to the change in speed rather than with the change in velocity?

 

[HomogenousCow]

The tangential acceleration cannot change the direction of the trajectory, the normal acceleration cannot change the magnitude of the trajectory.
All the tangential acceleration does is change the magnitude of the velocity, a.k.a speed.

 

[negation]

The tangential acceleration cannot change the direction of the trajectory, the normal acceleration cannot change the magnitude of the trajectory.
All the tangential acceleration does is change the magnitude of the velocity, a.k.a speed.

But hasn't it always been so that acceleration is associated with velocity rather than speed?

 

[HomogenousCow]

Speed is simply the magnitude of the velocity vector.
You might find this thread https://www.physicsforums.com/showthread.php?t=131904 useful.

 

[nasu]

But hasn't it always been so that acceleration is associated with velocity rather than speed?

The total acceleration is the derivative of velocity.
It's just that this total acceleration has no tangential component if the magnitude of the velocity is constant.

 

[jbriggs444]

As has been, accceleration is always the derivative of velocity.
In rotational motion (circular uniform), however, the tangential acceleration is the change in speed rather than velocity. Why is this so?

Usually you would talk about acceleration using cartesian coordinates -- x and y. But there is nothing magical about the choice of the x and y directions. Any pair of directions at 90 degree angles works just as well.

When you express acceleration as "tangential" and "radial", what you are doing, in essence, is adopting a coordinate system in which the x direction ("tangential") is [momentarily] lined up with the way the object is moving and the y direction ("radial") is [momentarily] on a line running from the chosen center point.

Caution: There some ambiguity here. Sometimes the object is not moving in a circular path around the chosen "center point". It might be traveling in an ellipse or a spiral. In such cases, one might be tempted to apply the term "tangential" to refer to the direction a circular path would take -- at right angles to the "radial" direction. Let us assume that our "tangential" is exactly lined up with the object's motion and that the "center point" that we are using is at right angles to that. [If the object is moving in a circular path around the chosen center point then both of these assumptions will automatically be fulfilled]

The x component of acceleration is always equal to the rate of change of the x component of velocity. But with this particular choice of coordinates, speed is identically equal to the x component of velocity. So tangential acceleration is the same as the rate of change of speed.