Motion of a Particle
Particle is a term used to denote an object of point size. A system of particles which formed into appreciable size is termed as body. These terms may apply equally to the same object. The earth for example may be assumed as a particle in comparison with its orbit, whereas to an observer on the earth, it is a body with appreciable size. In general, a particle is an object whose size is so small in comparison to the size of its path.

Rectilinear Translation (Motion Along a Straight Line)

Motion with constant velocity (uniform motion)
$s = vt$

Motion with constant acceleration

$v_f = v_i + at$

$s = v_it + \frac{1}{2}at^2$

${v_f}^2 = {v_i}^2 + 2as$

Free-falling body

$v = gt$

$h = \frac{1}{2}gt^2$

$v^2 = 2gh$

Note: From motion with constant acceleration, set vi = 0, vf = v, s = h, and a = g to get the free-fall formulas.

Motion with variable acceleration

$a = \dfrac{dv}{dt}$

$v = \dfrac{ds}{dt}$

$v \, dv = a \, ds$

Where
$s$ = distance
$h$ = height
$v$ = velocity
$v_i$ = initial velocity
$v_f$ = final velocity
$a$ = acceleration
$g$ = acceleration due to gravity (g = 9.81 m/s2 in SI = 32.2 ft/s2 in English)
$t$ = time

Note:
• $a$ is positive (+) if $v$ is increasing (accelerate).
• $a$ is negative (-) if $v$ is decreasing (decelerate).
• $g$ is positive (+) if the particle is moving downward.
• $g$ is negative (-)if the particle is moving upward.

Useful conversion factors:

 From To Multiply by Kilometers per hour (kph) Meter per second (m/sec) 1 / 3.6 Meter per second (m/sec) Kilometers per hour (kph or km/hr) 3.6 Miles per hour (mph) Feet per second (fps or ft/sec) 22 / 15 Feet per second (ft/sec) Miles per hour (mph or mi/hr) 15 / 22

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