Velocity of an Object

Average velocity measures the total distance covered over a time period, calculated by dividing position change by elapsed time. It describes motion between two moments but not at a specific instant. Instantaneous velocity, by contrast, captures the exact speed at one precise moment in time, requiring a limiting process rather than simple division.

Instantaneous velocity is found by calculating average velocities over progressively smaller time intervals surrounding a specific moment. As these intervals shrink toward zero, the average velocity values approach a single consistent number. This limiting value—what the average velocities approach but never exceed or fall short of—is the instantaneous velocity at that exact moment.

Instantaneous velocity describes motion at a single point in time with no duration. Since traditional velocity calculation requires dividing change in position by time interval, and no time interval exists at an instant, direct calculation is impossible. Instead, the limiting behavior of average velocities over shrinking intervals reveals the instantaneous velocity.

The tangent to a curve is a line that touches the position curve at exactly one point without crossing it. The slope of this tangent line equals the limit of average velocities as the time interval approaches zero, geometrically representing instantaneous velocity and providing a visual interpretation of the limiting process.

As the time interval becomes shorter, the calculated average velocity increasingly approximates the behavior at a single moment. The smaller the interval, the closer the average velocity approaches the true instantaneous velocity. This progressive refinement through interval reduction is the foundation of the limiting process that defines instantaneous velocity.

The slope of the line connecting two positions represents average velocity, showing how quickly position changes relative to time. This line spans two moments in time and provides a general sense of motion between those points. The slope is calculated as the change in position divided by the change in time over that interval.

As time intervals shrink around a specific moment, average velocity calculations increasingly reflect the motion at that precise instant rather than over a broader span. The values converge because they are measuring motion in an ever-narrower window, eventually approaching the true instantaneous velocity. This convergence defines the limit and establishes instantaneous velocity as a well-defined value.