Position Velocity Acceleration

Understanding Motion: Position, Velocity, and Acceleration

For an object moving along a straight line, position $s(t)$ describes its location at time $t$. Velocity $v(t) = s'(t)$ is the signed rate of change of position—positive indicates forward motion, negative indicates backward motion. Acceleration $a(t) = v'(t) = s''(t)$ measures how quickly velocity itself is changing. Note that speed, $|v(t)|$, differs from velocity: it measures how fast the object moves without directional information.

When Does Direction Change?

An object changes direction when velocity changes sign, typically where $v(t) = 0$. To identify these turning points, solve $s'(t) = 0$ and verify that velocity actually switches sign at each solution (it may only touch zero without reversing).

Speeding Up vs. Slowing Down

A common misconception is that negative acceleration always means slowing down. In reality, whether an object speeds up or slows down depends on whether velocity and acceleration have the same sign or opposite signs:

• Same sign (both positive or both negative): speed is increasing
• Opposite signs: speed is decreasing

For example, an object moving backward ($v < 0$) with negative acceleration ($a < 0$) is actually speeding up, since both indicate the same direction of change.

Displacement vs. Distance

Net displacement is simply $s(b) - s(a)$—the net change in position. However, total distance traveled must account for direction reversals. To find it, identify all times where $v(t) = 0$ within the interval, split the path into segments between these points, and sum the absolute position changes:

$\text{Total distance} = \sum_i |s(t_{i+1}) - s(t_i)|$

This distinction matters because backward motion adds to total distance rather than canceling it out algebraically.