What vector quantity is the time derivative of velocity?

Acceleration is the vector quantity defined as the time derivative of velocity.

In calculus notation, acceleration is written as a = dv/dt. Because velocity has both magnitude and direction, acceleration can describe a change in speed, a change in direction, or both. Its SI unit is the metre per second squared, which means the velocity changes by a certain number of metres per second every second.

This explains why an object moving at constant speed in a circle is still accelerating: its velocity direction continually changes. In straight-line motion, by contrast, acceleration usually appears as speeding up or slowing down. The second derivative of position with respect to time gives the same quantity, since velocity is the first derivative of position.

Acceleration is also central to Newton’s second law. For an object with constant mass, the net force equals mass multiplied by acceleration. A force therefore changes an object’s motion, while the acceleration measures the resulting change in velocity. The time derivative of momentum is force, which is a related but distinct relationship.