Shouldn't the right expression give us a scalar value, hence providing the tangential speed and not the tangential velocity?

If \(\vec{v}\) is always changing around a circle, how can we find the right expression to be a constant value? I was told that the velocity here refers to the instantaneous velocity, but I don't understand why we are using the whole circumference of the circle as the displacement.

What does it mean for displacement to be a circumference? Shouldn't that be considered distance instead?

1) The expression for tangential speed does indeed give us a scalar value, as it represents the magnitude of the velocity vector.
2) Although the velocity vector (vecv) is always changing in direction as an object moves along a circular path, the magnitude (speed) of the velocity vector can remain constant if the object is moving at a constant speed.
3) In this context, when we talk about the "velocity" referring to instantaneous velocity, we are considering the velocity of the object at a particular moment in time. Instantaneous velocity represents the velocity of an object at a specific point on its path, taking into account both the magnitude and direction of motion.

Explanation:

The tangential speed tells us how fast an object is moving along its circular path, without indicating the direction of motion. On the other hand, tangential velocity is a vector quantity that includes both the magnitude (speed) and direction of motion. So, you are correct that the expression gives us the tangential speed, not the tangential velocity.

This is because velocity is defined as the rate of change of displacement over time. In a circular motion, the object repeats its path over time, resulting in a constant displacement magnitude. Therefore, even though the direction of the velocity vector changes, the magnitude remains constant if the speed is constant.

The use of the entire circumference of the circle as the displacement is based on the fact that the object completes a full revolution around the circle. Displacement is a vector quantity that represents the change in position of an object, taking into account both the magnitude and direction of the change. Since the object returns to its starting point after completing a full revolution, the displacement is equal to the circumference of the circle.