Summary

• Frame of reference: Motion is described relative to a chosen origin and axes. Position is a vector $\vec{x}$; in one dimension, it is simply $x$ with sign.
• Path length & displacement: Distance is the total length of the path traveled (scalar). Displacement $\Delta x = x_f - x_i$ is the straight-line change in position (vector). Displacement can be zero even if distance is not.
• Average velocity & speed: Average velocity $\bar{v} = \frac{\Delta x}{\Delta t}$ (vector). Average speed $= \frac{\text{total distance}}{\text{total time}}$ (scalar). For a round trip, average velocity is zero but average speed is not.
• Instantaneous velocity: $v = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt}$. It is the slope of the $x$–$t$ graph at that instant.
• Acceleration: Average acceleration $\bar{a} = \frac{\Delta v}{\Delta t}$; instantaneous acceleration $a = \frac{dv}{dt} = \frac{d^2x}{dt^2}$. Acceleration is the slope of the $v$–$t$ graph.
• Equations of motion (constant acceleration $a$): For motion along a straight line with uniform $a$:

where $u$ = initial velocity, $v$ = final velocity, $s$ = displacement, $t$ = time.

• Graphical interpretation: Slope of $x$–$t$ gives $v$; slope of $v$–$t$ gives $a$; area under $v$–$t$ graph gives displacement.
• Free fall: A special case of constant acceleration ($a = g \approx 9.8\ \text{m/s}^2$ downward). All equations of motion apply with $a = -g$ (if upward is positive).

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Table of Physical Quantities