$v = dx/dt$, $a = dv/dt = d^2x/dt^2 = v(dv/dx)$. When relation is $t = f(x)$: $v = dx/dt = 1/(dt/dx) = 1/f'(x)$. Acceleration: $a = v(dv/dx)$. Compute $dv/dx$ from expression for $v$ in terms of $x$, multiply by $v$. This approach avoids solving for $x(t)$ explicitly.

For constant acceleration $a$: $v = u + at$. $s = ut + \frac{1}{2}at^2$. $v^2 = u^2 + 2as$. $s_n = u + a(2n-1)/2$ (displacement in $n$th second). These equations apply only when acceleration is constant. For variable acceleration: use calculus ($a = dv/dt$ or $a = v(dv/dx)$).

Horizontal: $x = u\cos\theta \cdot t$ (constant velocity). Vertical: $y = u\sin\theta \cdot t - \frac{1}{2}gt^2$. Time of flight: $T = 2u\sin\theta/g$. Maximum height: $H = u^2\sin^2\theta/(2g)$. Range: $R = u^2\sin 2\theta/g$. Maximum range at $\theta = 45°$: $R_{max} = u^2/g$. Equation of trajectory: $y = x\tan\theta - gx^2/(2u^2\cos^2\theta)$ (parabola).

Relative velocity of A with respect to B: $\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$. For two objects moving in same direction: relative speed $= |v_A - v_B|$. Opposite directions: relative speed $= v_A + v_B$. River-boat problem: boat velocity relative to ground $= \vec{v}_{boat} + \vec{v}_{river}$. Minimum drift: boat aims upstream. Minimum time: aim perpendicular to bank. Rain problem: $\vec{v}_{rain/man} = \vec{v}_{rain} - \vec{v}_{man}$.

Angular velocity $\omega = d\theta/dt = v/r$. Angular acceleration $\alpha = d\omega/dt$. Centripetal acceleration $a_c = v^2/r = \omega^2 r$. Centripetal force $F_c = mv^2/r$ (directed toward centre). In non-uniform circular motion: tangential acceleration $a_t = r\alpha$ (changes speed). Net acceleration $= \sqrt{a_c^2 + a_t^2}$. Conical pendulum, banked road: applications of centripetal force analysis.

When $a = f(t)$: integrate to get $v(t)$, integrate again for $x(t)$. When $a = f(v)$: $a = dv/dt \Rightarrow dt = dv/a$, integrate for $v(t)$. When $a = f(x)$: $a = v(dv/dx) \Rightarrow v\,dv = a\,dx$, integrate for $v(x)$. When $x = f(t)$ given directly: differentiate for $v$, differentiate again for $a$. When $t = f(x)$: use $v = 1/(dt/dx)$ and $a = v(dv/dx)$.

$x$-$t$ graph: slope $=$ velocity. Curvature indicates acceleration. $v$-$t$ graph: slope $=$ acceleration. Area under curve $=$ displacement. $a$-$t$ graph: area $=$ change in velocity. For uniform acceleration: $x$-$t$ is parabola, $v$-$t$ is straight line, $a$-$t$ is horizontal line. For uniform circular motion: $x$-$t$ and $v$-$t$ are sinusoidal, $|a|$ is constant.

First law: inertia. Object at rest stays at rest unless net force acts. Second law: $\vec{F} = m\vec{a}$. Third law: every action has equal and opposite reaction (on different objects). Free body diagram: isolate object, draw all forces acting ON it. Common forces: weight ($mg$ downward), normal ($N$ perpendicular to surface), friction ($\mu N$ opposing motion), tension ($T$ along string). Constraint equations: for connected bodies, relate their accelerations.