Physics · Ch 2 — Motion in a Straight Line

Summary

Frame of reference: an origin, axes and a clock relative to which motion is described. A frame is inertial if Newton's first law holds in it (unaccelerated); an accelerating frame is non-inertial and requires pseudo-forces to apply Newton's laws.
Path length (distance): the total length of the actual route travelled; a scalar, always $\ge 0$ .
Displacement: $\Delta x = x_2 - x_1$ , the net change in position; a vector along the line, and always $|\Delta x| \le$ path length.
Average speed = total path length / total time (scalar); average velocity $\bar v = \Delta x/\Delta t$ (vector). Average speed $\ge |\bar v|$ always, with equality only for strictly one-directional motion.
Instantaneous velocity $v = \lim_{\Delta t \to 0} \Delta x/\Delta t = dx/dt$ — the slope of the tangent to the x-t graph at an instant. Instantaneous speed $= |v|$ .
Average acceleration $\bar a = \Delta v/\Delta t$ ; instantaneous acceleration $a = dv/dt = d^2x/dt^2$ . Uniform acceleration means $a$ is constant; if $a$ is opposite to $v$ , the body decelerates (retardation).
Position-time graph: horizontal line $\Rightarrow$ rest; straight line of constant slope $\Rightarrow$ uniform velocity; parabola $\Rightarrow$ uniformly accelerated motion (slope of tangent = instantaneous velocity).
Velocity-time graph: slope = acceleration; area under the graph = displacement (from $\int v\,dt$ ). …