Kinematics is the branch of mechanics that describes the motion of objects — their position, displacement, velocity, and acceleration — without considering the forces that cause the motion.

Scope (What Kinematics Ignores)

• Does not analyze forces, energy, or momentum.
• Only describes how things move, not why.

Core Quantities

• Position: \[ \overrightarrow{r}(t) \] locates a particle in space.
• Displacement: change in position \[ \Delta \overrightarrow{r} = \overrightarrow{r}_2 - \overrightarrow{r}_1 \]
• Velocity: rate of change of position \[ \overrightarrow{v}_{\text{avg}} = \frac{\Delta \overrightarrow{r}}{\Delta t}, \quad \overrightarrow{v} = \frac{d\overrightarrow{r}}{dt} \]
• Acceleration: rate of change of velocity \[ \overrightarrow{a}_{\text{avg}} = \frac{\Delta \overrightarrow{v}}{\Delta t}, \quad \overrightarrow{a} = \frac{d\overrightarrow{v}}{dt} = \frac{d^2 \overrightarrow{r}}{dt^2} \]

1D Motion (Straight Line)

For constant acceleration (\(a=\) constant), the kinematic (“SUVAT”) equations are:

\[ v = u + a t,\qquad s = ut + \tfrac{1}{2} a t^2,\qquad v^2 = u^2 + 2 a s \]

where \(u\) = initial velocity, \(v\) = final velocity, \(a\) = acceleration, \(t\) = time, \(s\) = displacement along the line.

2D Motion (Projectiles)

With initial speed \(u\) at angle \(\theta\) above horizontal (no air resistance):

\[ u_x = u\cos\theta,\quad u_y = u\sin\theta,\quad a_x = 0,\ a_y = -g \] \[ \text{Range: } R = \frac{u^2\sin 2\theta}{g},\quad \text{Max height: } H = \frac{u^2\sin^2\theta}{2g},\quad \text{Time of flight: } T = \frac{2u\sin\theta}{g} \]

Relative Motion (Galilean)

Velocities add/subtract as vectors: \[ \overrightarrow{v}_{A/B} = \overrightarrow{v}_{A/E} - \overrightarrow{v}_{B/E} \] Example: rain appears slanted because \[ \overrightarrow{v}_{\text{rain/you}} = \overrightarrow{v}_{\text{rain/ground}} -\overrightarrow{v}_{\text{you/ground}} \]

Uniform Circular Motion (Kinematics View)

\[ \omega = \frac{d\theta}{dt},\quad \alpha = \frac{d\omega}{dt},\quad v = r\omega,\quad a_t = r\alpha,\quad a_c = \frac{v^2}{r} = r\omega^2 \ (\text{toward center}) \]

Graphs You Should Read

• x–t graph: slope = \(v\). Straight line ⇒ constant \(v\); curve ⇒ changing \(v\).
• v–t graph: slope = \(a\); area under curve = displacement \(s\).
• a–t graph: area under curve = change in velocity \(\Delta v\).

Units & Dimensions

• Displacement/Position: \([L]\) (metre, m)
• Velocity: \([LT^{-1}]\) (m/s)
• Acceleration: \([LT^{-2}]\) (m/s\(^2\))

Typical Pitfalls

• Confusing distance (scalar) with displacement (vector).
• Applying \(v^2 = u^2 + 2as\) when \(a\) is not constant.
• Mixing signs in 1D (choose positive direction once and stick to it).
• For projectiles, treating \(g\) as positive upward—use \(a_y=-g\) if up is +ve.

Mini Worked Example

A car starts from rest and accelerates uniformly at \(2\,\mathrm{m\,s^{-2}}\) for \(6\,\mathrm{s}\). Find \(v\) and \(s\).

\[ v = u + at = 0 + 2\cdot 6 = 12\ \mathrm{m/s},\qquad s = ut + \tfrac{1}{2}at^2 = 0 + \tfrac{1}{2}\cdot 2 \cdot 36 = 36\ \mathrm{m}. \]