Motion in a Straight Line

Position: location of object w.r.t. reference point (origin). Path length (distance): total length of actual path — scalar, always positive. Displacement: change in position = final - initial position — vector, can be +ve, -ve or zero. Example: Run 4m east then 3m west: path=7m, displacement=1m east.

Average velocity = total displacement / total time. Average speed = total distance / total time. These can be different! Instantaneous velocity = velocity at a specific instant = limit of average velocity as Δt→0 = dx/dt (derivative of position w.r.t. time). Speedometer shows instantaneous speed.

Acceleration a = dv/dt = rate of change of velocity. Uniform acceleration: same acceleration throughout (free fall, object on inclined plane). Non-uniform: acceleration changes (car in traffic). Positive acceleration: speeding up in +ve direction. Negative acceleration (retardation): slowing down.

For uniform acceleration: v=u+at, s=ut+½at², v²=u²+2as, s_nth=u+a(2n-1)/2. x-t graph: slope=velocity. v-t graph: slope=acceleration, area=displacement. a-t graph: area=change in velocity. These equations ONLY apply when acceleration is constant.