Source: High school physics (Chinese)
Problem
An object is in uniform circular motion (UCM).
- Is its velocity constant?
- Is its acceleration constant?
[Q1] No. The velocity is not constant because its direction is continuously changing, even though its magnitude (speed) is constant. [Q2] No. The acceleration is not constant because its direction is continuously changing (always pointing toward the center of the circle), even though its magnitude is constant.
[Q1] Is its velocity constant?
Velocity, $\vec{v}$, is a vector quantity defined by its magnitude (speed) and direction. In uniform circular motion (UCM), the speed, $v = |\vec{v}|$, is constant by definition. However, the direction of motion is always tangent to the circular path. As the object moves along the circle, this tangential direction continuously changes. Since the direction of the velocity vector changes, the velocity vector $\vec{v}$ is not constant.
[Q2] Is its acceleration constant?
Acceleration, $\vec{a}$, is the rate of change of the velocity vector ($\vec{a} = \frac{\Delta \vec{v}}{\Delta t}$). Since the velocity $\vec{v}$ is changing (as established in Q1), there must be a non-zero acceleration. In UCM, this acceleration is the centripetal acceleration, $\vec{a}_c$, which is responsible for changing the direction of the velocity.
The magnitude of the centripetal acceleration is given by:
$$a_c = \frac{v^2}{r}$$where $v$ is the constant speed and $r$ is the constant radius of the circle. Therefore, the magnitude of the acceleration is constant.
The direction of the centripetal acceleration is always radially inward, pointing from the object towards the center of the circle. As the object revolves around the circle, this direction continuously changes. For example, at the top of a vertical circle, the acceleration points downwards, while at the bottom, it points upwards. Since the direction of the acceleration vector changes, the acceleration vector $\vec{a}$ is not constant.