Kinematics Compendium

Kinematics — From First Principles to Advanced Forms

Base SUVAT, projectiles, polar/cylindrical/spherical, Frenet–Serret, relative & rotating frames, SHM, rolling constraints, jerk/snap, and drag solutions.

0) The bedrock (definitions & operators)

  • Position: $\mathbf r(t)$ in $\mathbb R^2$ or $\mathbb R^3$.
  • Displacement: $\Delta\mathbf r=\mathbf r(t_2)-\mathbf r(t_1)$.
  • Average velocity: $\overline{\mathbf v}=\Delta\mathbf r/\Delta t$.
  • Instantaneous velocity: $\displaystyle \mathbf v=\frac{d\mathbf r}{dt}$. Speed $v=|\mathbf v|$.
  • Instantaneous acceleration: $\displaystyle \mathbf a=\frac{d\mathbf v}{dt}=\frac{d^2\mathbf r}{dt^2}$.
  • Higher derivatives: jerk $\mathbf j=\dfrac{d\mathbf a}{dt}$, snap $\mathbf s=\dfrac{d\mathbf j}{dt}$.
  • Arc length: $s(t)=\int_{t_0}^{t} |\mathbf v(\tau)|\,d\tau,\; \dfrac{ds}{dt}=v$.
  • Re-parameterization: if $\mathbf r=\mathbf r(s)$ then $\dot{\mathbf r}=\mathbf T\,v$, unit tangent $\mathbf T=d\mathbf r/ds$.

1) Constant acceleration in 1-D (the SUVAT family)

Assume $a=$ const along line $x$:

$$ v=v_0+at,\quad x=x_0+v_0 t+\tfrac12 a t^2,\quad v^2=v_0^2+2a\,\Delta x,\quad \Delta x=\tfrac{(v+v_0)}{2}t. $$

Derivation: integrate $\dot v=a\Rightarrow v=v_0+at$, then $\dot x=v\Rightarrow x=x_0+\int(v_0+at)\,dt$, eliminate $t$ for the $v^2$ form.

Domain: valid only for constant $a$. Otherwise use integrals or piecewise segments.

2) Projectile motion (no air resistance)

Launch $(x_0,y_0)$ at speed $v_0$ and angle $\theta$, with gravity $g$ downward.

$x(t)=x_0+v_0\cos\theta\,t$, $y(t)=y_0+v_0\sin\theta\,t-\tfrac12 g t^2$.

  • Level ground $(y_0=0\to y=0)$: $t_f=\dfrac{2v_0\sin\theta}{g}$; $R=\dfrac{v_0^2\sin2\theta}{g}$; $H=\dfrac{v_0^2\sin^2\theta}{2g}$.
  • Different launch height $y_0$: $t_f=\dfrac{v_0\sin\theta+\sqrt{v_0^2\sin^2\theta+2gy_0}}{g}$, $R=v_0\cos\theta\,t_f$.
  • On an incline (slope $\phi$): $R_{\text{slope}}=\dfrac{2v_0^2\cos\theta\,\sin(\theta-\phi)}{g\cos^2\phi}$.

3) Vector kinematics in plane polar form $(r,\theta)$

Unit vectors rotate: $\dot{\mathbf e}_r=\dot\theta\,\mathbf e_\theta$, $\dot{\mathbf e}_\theta=-\dot\theta\,\mathbf e_r$.

$$ \mathbf v=\dot r\,\mathbf e_r + r\dot\theta\,\mathbf e_\theta,\qquad \mathbf a=(\ddot r-r\dot\theta^2)\,\mathbf e_r + (r\ddot\theta+2\dot r\dot\theta)\,\mathbf e_\theta. $$

4) Frenet–Serret (TNB) frame & curvature

Unit tangent $\mathbf T$, normal $\mathbf N$, binormal $\mathbf B=\mathbf T\times\mathbf N$. Curvature $\kappa=|d\mathbf T/ds|$, radius $\rho=1/\kappa$.

Acceleration decomposition:

$$ \mathbf a=\underbrace{\dot v}_{a_t}\mathbf T+\underbrace{\frac{v^2}{\rho}}_{a_n}\mathbf N. $$

Torsion $\tau$ satisfies the Frenet–Serret system:

$$ \frac{d\mathbf T}{ds}=\kappa\mathbf N,\quad \frac{d\mathbf N}{ds}=-\kappa\mathbf T+\tau\mathbf B,\quad \frac{d\mathbf B}{ds}=-\tau\mathbf N. $$

5) Circular & rotational kinematics

For radius $r$: $v=\omega r$, $a_n=\omega^2 r$, $a_t=\alpha r$.

Constant $\alpha$: $\omega=\omega_0+\alpha t$, $\theta=\theta_0+\omega_0 t+\tfrac12\alpha t^2$, $\omega^2=\omega_0^2+2\alpha\,\Delta\theta$.

Rigid body: $\mathbf v(\mathbf r)=\boldsymbol\omega\times\mathbf r$, $\mathbf a(\mathbf r)=\boldsymbol\alpha\times\mathbf r+\boldsymbol\omega\times(\boldsymbol\omega\times\mathbf r)$.

6) Relative motion (Galilean)

Frames with $\mathbf r'=\mathbf r-\mathbf V t$ (constant $\mathbf V$): $\mathbf v'=\mathbf v-\mathbf V$, $\mathbf a'=\mathbf a$.

7) Rotating reference frames (transport theorem)

For any vector $\mathbf q$: $(d\mathbf q/dt)_A=(d\mathbf q/dt)_B+\boldsymbol\omega\times\mathbf q$.

Point kinematics:

$$ \mathbf v_P=\mathbf v_O+\boldsymbol\omega\times\mathbf r+\mathbf v_{\text{rel}}, \qquad \mathbf a_P=\mathbf a_O+\boldsymbol\alpha\times\mathbf r+\boldsymbol\omega\times(\boldsymbol\omega\times\mathbf r)+2\boldsymbol\omega\times\mathbf v_{\text{rel}}+\left(\frac{d\mathbf v_{\text{rel}}}{dt}\right)_B. $$

8) Cylindrical & spherical components

Cylindrical $(\rho,\phi,z)$:

$$ \mathbf v=\dot\rho\,\mathbf e_\rho+\rho\dot\phi\,\mathbf e_\phi+\dot z\,\mathbf e_z,\qquad \mathbf a=(\ddot\rho-\rho\dot\phi^2)\mathbf e_\rho+(\rho\ddot\phi+2\dot\rho\dot\phi)\mathbf e_\phi+\ddot z\,\mathbf e_z. $$

Spherical $(r,\theta,\varphi)$:

$$ \mathbf v=\dot r\,\mathbf e_r + r\dot\theta\,\mathbf e_\theta + r\sin\theta\,\dot\varphi\,\mathbf e_\varphi, $$ $$ \begin{aligned} \mathbf a={}&(\ddot r-r\dot\theta^2-r\sin^2\theta\,\dot\varphi^2)\,\mathbf e_r \\ &+ (r\ddot\theta+2\dot r\dot\theta - r\sin\theta\cos\theta\,\dot\varphi^2)\,\mathbf e_\theta \\ &+ (r\sin\theta\,\ddot\varphi + 2\dot r\sin\theta\,\dot\varphi + 2r\dot\theta\cos\theta\,\dot\varphi)\,\mathbf e_\varphi. \end{aligned} $$

9) Rolling without slipping

Constraint at contact: $v_{\text{contact}}=0\Rightarrow v_{\text{COM}}=\omega R$. For any rim point $P$, $\mathbf v_P=\mathbf v_{\text{COM}}+\boldsymbol\omega\times\mathbf r_{P/\text{COM}}$.

10) Simple harmonic motion (SHM)

$x(t)=A\cos(\omega t+\phi)\Rightarrow v=-A\omega\sin(\cdot),\; a=-\omega^2 x$. Period $T=2\pi/\omega$. SHM is a projection of uniform circular motion.

11) Non-constant acceleration

  • Time-dependent: if $a(t)=a_0+kt$ then $v=v_0+a_0 t+\tfrac12 k t^2$, $x=x_0+v_0 t+\tfrac12 a_0 t^2+\tfrac16 k t^3$.
  • State-dependent: if $a=a(x)$ then $v\,dv/dx=a(x)\Rightarrow \tfrac12 v^2=\int a(x)\,dx + C$.

12) Air resistance (closed forms)

Linear drag $F_d=bv$, terminal $v_T=mg/b$:

$$ v(t)=v_T(1-e^{-t/\tau}),\quad x(t)=v_T t - v_T \tau(1-e^{-t/\tau}),\quad \tau=m/b. $$

Quadratic drag $F_d=cv^2$, terminal $v_T=\sqrt{mg/c}$:

$$ v(t)=v_T\tanh\!\left(\frac{g t}{v_T}\right),\quad x(t)=\frac{v_T^2}{g}\,\ln\!\cosh\!\left(\frac{g t}{v_T}\right). $$

Horizontal components with quadratic drag couple; solve numerically or with small-drag approximations.

13) Orientation kinematics

Euler angles (sequence-dependent mapping), axis–angle $\boldsymbol\omega=\dot\theta\,\hat{\mathbf u}$, and quaternions $\dot q=\tfrac12[0,\boldsymbol\omega]\otimes q$ encode orientation rates (purely kinematic).

14) Dimensional analysis

$[x]=L$, $[v]=LT^{-1}$, $[a]=LT^{-2}$, $[\omega]=T^{-1}$, $[\alpha]=T^{-2}$. Example: projectile range $R=v_0^2\sin2\theta/g$ has units $L$.

15) Common pitfalls & scope

  • SUVAT needs constant $a$; with drag or variable $g$, switch methods.
  • Use radians for calculus with angles.
  • In polar/cylindrical/spherical, unit vectors are time-dependent.
  • “Centrifugal/Coriolis” terms here are kinematic in rotating frames; forces arise in dynamics.

16) Compact tables

Linear ↔ Rotational: $x\leftrightarrow\theta$, $v\leftrightarrow\omega$, $a\leftrightarrow\alpha$, $s=r\theta$, $v=r\omega$, $a_t=r\alpha$, $a_n=v^2/r=\omega^2 r$.

Planar polar: $\mathbf v=\dot r\,\mathbf e_r+r\dot\theta\,\mathbf e_\theta$, $\mathbf a=(\ddot r-r\dot\theta^2)\mathbf e_r+(r\ddot\theta+2\dot r\dot\theta)\mathbf e_\theta$.

Rigid point: $\mathbf v_B=\mathbf v_A+\boldsymbol\omega\times\mathbf r_{B/A}$, $\mathbf a_B=\mathbf a_A+\boldsymbol\alpha\times\mathbf r_{B/A}+\boldsymbol\omega\times(\boldsymbol\omega\times\mathbf r_{B/A})$.

17) Worked micro-examples

  1. Stop time: $v_0=28\,\text{m/s}$, $a=-7\,\text{m/s}^2$. $t=4\,\text{s}$, distance $\Delta x=56\,\text{m}$.
  2. Projectile from height $h=12\,\text{m}$, $v_0=20\,\text{m/s}$, $\theta=30^\circ$: $t_f\approx 2.85\,\text{s}$, $R\approx 49.3\,\text{m}$.
  3. Polar acceleration: $r=ct$, $\theta=\beta t$ $\Rightarrow \mathbf a=(-r\beta^2)\mathbf e_r+(2c\beta)\mathbf e_\theta$.