Position, Velocity, and Acceleration

For an object moving along a line with position $s(t)$:

Quantity	Formula	Meaning
velocity	$v(t) = s'(t)$	rate and direction of motion
acceleration	$a(t) = v'(t) = s''(t)$	how fast the velocity changes
speed	$\lvert v(t) \rvert$	how fast, ignoring direction

Reading the signs

• $v > 0$: moving right (or up). $v < 0$: moving left (or down). $v = 0$: at rest, possibly turning around.
• Speeding up when $v$ and $a$ have the same sign; slowing down when they have opposite signs. A car moving left ($v < 0$) with $a < 0$ is speeding up.

Example

$s(t) = t^3 - 6t^2 + 9t$ gives $v(t) = 3t^2 - 12t + 9 = 3(t - 1)(t - 3)$ and $a(t) = 6t - 12$.

The object moves right until $t = 1$, left until $t = 3$, then right again.

Displacement vs distance

Displacement is $s(\text{end}) - s(\text{start})$. Distance traveled adds up every leg, so split the trip where $v$ changes sign. Above, from $t = 0$ to $t = 3$: displacement $0$, but distance $4 + 4 = 8$.

Free fall

Near Earth's surface, height in feet is $h(t) = -16t^2 + v_0 t + h_0$ (in meters, $-4.9t^2 + v_0t + h_0$). The top of the flight is where $v(t) = 0$.