Kinematics of Particles Notes

Last reviewed 16 Sept 2026 · 5 min read

Kinematics

Kinematics describes motion — position, velocity and acceleration — without asking what forces cause it (that is kinetics). A particle is a body whose size does not matter for the motion being studied.

DefinitionBasic quantities

Distance — total path length (scalar). Displacement — change in position (vector).
Speed — rate of change of distance. Velocity $v = \frac{d x}{d t}$ (vector).
Acceleration $a = \frac{d v}{d t} = \frac{d ^{2} x}{d t ^{2}} = v \frac{d v}{d x}$ .
Average velocity = total displacement ÷ total time.

Rectilinear motion with uniform acceleration

FormulaEquations of motion (constant

a

)

v = u + a t

s = u t + \frac{1}{2} a t^{2}

v^{2} = u^{2} + 2 a s

s = \frac{u + v}{2} t

Distance covered in the $n$ th second: $s_{n} = u + \frac{a}{2} (2 n - 1)$

Motion under gravity

Take $g$ = 9.81 m/s² downward (neglecting air resistance).

Body thrown up with speed $u$ : time to reach the top $= u / g$ ; maximum height $= u^{2} /2 g$ ; total time of flight (back to the start) $= 2 u / g$ ; it returns with speed $u$ .
Body dropped from height $h$ : time $= 2 h / g$ ; striking velocity $= 2 g h$ .

Variable acceleration

When $a$ changes, integrate:

$a = f (t)$ : $v = \int a d t$ , $x = \int v d t$ .
$a = f (v)$ : $d t = \frac{d v}{f ( v )}$ or $d x = \frac{v d v}{f ( v )}$ .
$a = f (x)$ : $v d v = f (x) d x$ .

Motion graphs

Graph	Slope gives	Area gives
Displacement–time	Velocity	—
Velocity–time	Acceleration	Displacement
Acceleration–time	Rate of change of acceleration (jerk)	Change in velocity

Relative velocity

The velocity of A relative to B:

v_{A / B} = v_{A} - v_{B}

Classic problems: two trains, rain and umbrella angle, boat crossing a river.

Boat crossing a river of width $d$ , boat speed $v_{b}$ in still water, river speed $v_{r}$ :

Shortest time: head straight across; $t = d / v_{b}$ ; drift downstream $= v_{r} d / v_{b}$ .
Shortest path (straight across, $v_{b} > v_{r}$ ): head upstream at angle $θ$ with $sin θ = v_{r} / v_{b}$ to the perpendicular; $t = \frac{d}{v _{b}^{2} - v _{r}^{2}}$ .

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Kinematics

Rectilinear motion with uniform acceleration

Motion under gravity

Variable acceleration

Motion graphs

Relative velocity

This chapter is in the syllabus of