**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 acceleration

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

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

Free-falling body

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

$v^2 = 2gh$

Note: From motion with constant acceleration, set v_{i} = 0, v_{f} = v, s = h, and a = g to get the free-fall formulas.

Motion with variable acceleration

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

$v \, dv = a \, ds$

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/s^{2} in SI = 32.2 ft/s^{2} 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 |