1.3 Conceptual Analysis of Motion

Position, velocity, and acceleration relationships

Velocity is the rate of change of position, and acceleration is the rate of change of velocity. When velocity and acceleration have the same sign, an object speeds up; when they have opposite signs, it slows down. An object can have zero velocity but nonzero acceleration (momentarily at rest, like at the top of a toss).

Reading motion graphs

On a position-vs-time graph, the slope equals velocity. On a velocity-vs-time graph, the slope equals acceleration and the area under the curve equals displacement. A straight line means constant velocity or constant acceleration depending on the graph; curvature signals changing velocity (nonzero acceleration).

Interpreting motion conceptually

Motion can be analyzed qualitatively without equations: constant slope on a position graph means constant velocity, while a steepening slope means increasing speed. Comparing graphs of position, velocity, and acceleration for the same motion helps verify consistency — the velocity graph is the derivative-like trend of position, and acceleration is the same for velocity.

Key terms

Velocity (conceptual definition)

The rate of change of position with time; the slope of a position-vs-time graph.

Acceleration (conceptual definition)

The rate of change of velocity with time; the slope of a velocity-vs-time graph.

Same sign velocity & acceleration

Object is speeding up.

Opposite sign velocity & acceleration

Object is slowing down.

Area under velocity-vs-time graph

Equals the displacement of the object over that time interval.

Try an AP-style question

A ball is thrown straight up. At the very top, what are its velocity and acceleration?

1. v = g; a = 0
2. v = 0; a = g upward
3. v = 0; a = 0
4. v = 0; a = g downward