Physics fundamentals

2026-09-25

Formulas for first-year physics in one place: mechanics, waves, fluids, heat, electromagnetism, optics and pointers into modern physics, with SI units throughout. Constant values are in constants, units and conversions, symbols in notation, and the algebra in math fundamentals.

Units and dimensions

Every equation must balance in dimensions as well as numbers. Write the dimension of a quantity as $[\,\cdot\,]$ built from $\mathsf{M}$ (mass), $\mathsf{L}$ (length), $\mathsf{T}$ (time), $\mathsf{I}$ (current) and $\Theta$ (temperature). The SI base units and derived-unit definitions are on the constants and units sheet.

Quantity	Dimension	SI unit
velocity	$\mathsf{L\,T^{-1}}$	m/s
acceleration	$\mathsf{L\,T^{-2}}$	m/s²
force	$\mathsf{M\,L\,T^{-2}}$	N = kg·m/s²
energy, work	$\mathsf{M\,L^2\,T^{-2}}$	J = N·m
power	$\mathsf{M\,L^2\,T^{-3}}$	W = J/s
pressure	$\mathsf{M\,L^{-1}\,T^{-2}}$	Pa = N/m²
momentum	$\mathsf{M\,L\,T^{-1}}$	kg·m/s = N·s
charge	$\mathsf{I\,T}$	C = A·s
voltage	$\mathsf{M\,L^2\,T^{-3}\,I^{-1}}$	V = J/C
resistance	$\mathsf{M\,L^2\,T^{-3}\,I^{-2}}$	Ω = V/A
magnetic field	$\mathsf{M\,T^{-2}\,I^{-1}}$	T = N/(A·m)
angle, strain	dimensionless	rad, none

Check	Rule
adding, subtracting, $=$	every term has the same dimension
inside $\sin$ , $e^x$ , $\ln$	argument is dimensionless ( $\omega t$ , $t/\tau$ , $x/x_0$ )
guessing a law	$T = C\,L^a g^b$ with $[T] = \mathsf{T}$ forces $a = \tfrac12$ , $b = -\tfrac12$ : $T \propto \sqrt{L/g}$
units in calculations	convert to SI first (km/h to m/s: divide by $3.6$ )

Vectors quickly

Operation	Formula
components from polar	$v_x = v\cos\theta$ , $v_y = v\sin\theta$
magnitude, direction	$v = \sqrt{v_x^2 + v_y^2}$ , $\theta = \operatorname{atan2}(v_y, v_x)$
add, subtract	component-wise
dot product (scalar)	$\vec a \cdot \vec b = ab\cos\theta$ : work, flux, power
cross product (vector)	$\lVert \vec a \times \vec b \rVert = ab\sin\theta$ , right-hand rule: torque, Lorentz force

Which equation?

You have or want	Reach for	Section
constant acceleration, time not given	$v^2 = u^2 + 2as$	Kinematics
forces on a body, want acceleration	$\sum \vec F = m\vec a$ on a free-body diagram	Newton's laws
speeds at two heights, no time, no friction	$K_1 + U_1 = K_2 + U_2$	Energy
collision or explosion	$\sum \vec p$ conserved	Momentum
spinning body under a torque	$\tau = I\alpha$	Rotation
spinning body changes shape	$I_1\omega_1 = I_2\omega_2$	Rotation
orbit period against radius	$T^2 = 4\pi^2 r^3 / GM$	Gravitation
mass on a spring or pendulum period	$T = 2\pi\sqrt{m/k}$ , $T = 2\pi\sqrt{L/g}$	Osc. and waves
wave speed, frequency, wavelength	$v = f\lambda$	Osc. and waves
moving source or observer changes pitch	Doppler	Osc. and waves
pressure at a depth	$p = p_0 + \rho g h$	Fluids
floating or sinking	$F_b = \rho_\text{fluid} V_\text{sub} g$	Fluids
flow speeds up in a narrowing pipe	$A_1v_1 = A_2v_2$ then Bernoulli	Fluids
heating or melting a substance	$Q = mc\Delta T$ , $Q = mL$	Thermodynamics
gas pressure, volume and temperature	$pV = nRT$	Thermodynamics
best possible engine efficiency	$\eta_C = 1 - T_c/T_h$	Thermodynamics
current in a circuit	$V = IR$ plus Kirchhoff's rules	E&M
capacitor voltage over time	$V_C = V_0(1 - e^{-t/RC})$	E&M
charged particle in a magnetic field	$r = mv/qB$	E&M
changing flux, want voltage	$\mathcal{E} = -N\,d\Phi_B/dt$	E&M
light crossing a boundary	$n_1\sin\theta_1 = n_2\sin\theta_2$	Optics
where a lens puts the image	$1/f = 1/d_o + 1/d_i$	Optics
photon energy, particle wavelength	$E = hf$ , $\lambda = h/p$	Modern physics

Kinematics

$v = \dfrac{dx}{dt}$ , $a = \dfrac{dv}{dt}$ ; displacement is the area under $v(t)$ , change in velocity the area under $a(t)$ .

Constant acceleration (SUVAT)

$s$ displacement, $u$ initial velocity, $v$ final velocity, $a$ acceleration, $t$ time. Pick the equation that leaves out the variable you neither know nor want.

Equation	Missing
$v = u + at$	$s$
$s = ut + \tfrac12 at^2$	$v$
$s = vt - \tfrac12 at^2$	$u$
$v^2 = u^2 + 2as$	$t$
$s = \tfrac12(u + v)\,t$	$a$

Free fall: $a = -g$ with $g = 9.81\ \text{m/s}^2$ (up positive).

Projectile (no air resistance)

Horizontal and vertical motion are independent; only $y$ accelerates.

x = v_0\cos\theta\,t, \qquad y = v_0\sin\theta\,t - \tfrac12 g t^2, \qquad y = x\tan\theta - \frac{g x^2}{2 v_0^2\cos^2\theta}

Level ground result	Formula
time of flight	$T = \dfrac{2v_0\sin\theta}{g}$
maximum height	$H = \dfrac{v_0^2\sin^2\theta}{2g}$
range	$R = \dfrac{v_0^2\sin 2\theta}{g}$ , largest at $45^\circ$
speed at apex	$v_0\cos\theta$ (vertical part is zero)

Relative velocity: $\vec v_{A/C} = \vec v_{A/B} + \vec v_{B/C}$ .

Newton's laws and forces

Law	Statement
first	no net force: velocity constant (inertial frames)
second	$\sum \vec F = m\vec a = \dfrac{d\vec p}{dt}$
third	$\vec F_{AB} = -\vec F_{BA}$ : equal, opposite, on different bodies

Free-body diagrams

Isolate one body; draw it as a dot or box.
Draw every force on it: weight, normal, tension, friction, spring, drag, applied.
Choose axes along the acceleration (along an incline: $x$ down the slope).
Write $\sum F_x = ma_x$ and $\sum F_y = ma_y$ ; solve.
Several bodies: one diagram each, linked by shared tension or contact forces.

Common forces

Force	Formula	Direction
weight	$W = mg$	down
normal	from $\sum F_\perp = 0$ ( $mg\cos\theta$ on a slope)	perpendicular to the surface
static friction	$f_s \le \mu_s N$	opposes impending slip
kinetic friction	$f_k = \mu_k N$	opposes sliding
spring (Hooke)	$F = -kx$	toward equilibrium
drag (fast)	$F_D = \tfrac12 \rho C_D A v^2$	against velocity
centripetal	$F_c = \dfrac{mv^2}{r} = m\omega^2 r$	toward the center

On an incline of angle $\theta$ : along the slope $mg\sin\theta$ , into it $mg\cos\theta$ ; sliding with friction gives $a = g(\sin\theta - \mu_k\cos\theta)$ . Springs in series: $1/k = \sum 1/k_i$ ; in parallel: $k = \sum k_i$ .

Circular motion

Quantity	Formula
speed, period	$v = \omega r = \dfrac{2\pi r}{T}$
centripetal accel.	$a_c = \dfrac{v^2}{r} = \omega^2 r$
flat curve, friction	$v_\text{max} = \sqrt{\mu_s g r}$
banked curve, no friction	$\tan\theta = \dfrac{v^2}{rg}$
top of vertical loop	$v_\text{min} = \sqrt{gr}$

Centripetal force is not a new force: it is the net inward part of the forces already on the diagram.

Work, energy and power

Quantity	Formula	Unit
work (constant force)	$W = \vec F \cdot \vec d = Fd\cos\theta$	J
work (varying force)	$W = \int \vec F \cdot d\vec r$	J
kinetic energy	$K = \tfrac12 mv^2 = \dfrac{p^2}{2m}$	J
gravitational PE (near surface)	$U_g = mgh$	J
elastic PE	$U_s = \tfrac12 kx^2$	J
work-energy theorem	$W_\text{net} = \Delta K$	J
force from potential	$F_x = -\dfrac{dU}{dx}$	N
power	$P = \dfrac{dW}{dt} = \vec F \cdot \vec v$	W
efficiency	$\eta = P_\text{out}/P_\text{in}$	none

Conservation of energy, with $W_\text{nc}$ the work by non-conservative forces (friction, drag, a motor):

K_1 + U_1 + W_\text{nc} = K_2 + U_2

Friction on a flat path: $W_\text{nc} = -\mu_k N d$ . A drop from rest through height $h$ : $v = \sqrt{2gh}$ .

Momentum and collisions

Quantity	Formula
momentum	$\vec p = m\vec v$
impulse	$\vec J = \int \vec F\,dt = \vec F_\text{avg}\Delta t = \Delta \vec p$
conservation	no external net force: $\sum \vec p_\text{before} = \sum \vec p_\text{after}$
center of mass	$\vec r_\text{cm} = \dfrac{\sum m_i \vec r_i}{\sum m_i}$ , $\ \vec F_\text{ext} = M\vec a_\text{cm}$

Collision	Momentum	Kinetic energy	1D result ( $m_2$ at rest)
elastic	conserved	conserved	$v_1' = \dfrac{m_1 - m_2}{m_1 + m_2}v_1$ , $\ v_2' = \dfrac{2m_1}{m_1 + m_2}v_1$
inelastic	conserved	some lost	need one more fact, such as the coefficient of restitution
perfectly inelastic	conserved	most lost	stick together: $v' = \dfrac{m_1 v_1}{m_1 + m_2}$

Coefficient of restitution: $e = \dfrac{v_2' - v_1'}{v_1 - v_2}$ ( $1$ elastic, $0$ stuck).

Rotation

Linear	Rotational	Link
$x$ (m)	$\theta$ (rad)	arc $s = r\theta$
$v$	$\omega$ (rad/s)	$v = \omega r$
$a$	$\alpha$ (rad/s²)	$a_t = \alpha r$
$m$	$I$ (kg·m²)	$I = \sum m_i r_i^2$
$F$	$\tau$ (N·m)	$\vec\tau = \vec r \times \vec F$ , $\ \tau = rF\sin\phi$
$F = ma$	$\tau_\text{net} = I\alpha$
$p = mv$	$L = I\omega$ (kg·m²/s)	$\vec L = \vec r \times \vec p$
$K = \tfrac12 mv^2$	$K = \tfrac12 I\omega^2$
$P = Fv$	$P = \tau\omega$

SUVAT carries over with $\theta, \omega_0, \omega, \alpha, t$ . Static equilibrium: $\sum \vec F = 0$ and $\sum \vec \tau = 0$ about any point. Rolling without slipping: $v = \omega R$ , $K = \tfrac12 mv^2 + \tfrac12 I\omega^2$ . With no external torque, $L$ is conserved.

Moments of inertia

Body (mass $M$ )	Axis	$I$
point mass at distance $r$	any	$Mr^2$
thin ring or thin-walled cylinder	central axis	$MR^2$
solid disc or cylinder	central axis	$\tfrac12 MR^2$
thick-walled cylinder	central axis	$\tfrac12 M(R_1^2 + R_2^2)$
solid sphere	diameter	$\tfrac25 MR^2$
thin spherical shell	diameter	$\tfrac23 MR^2$
thin rod, length $L$	center, perpendicular	$\tfrac1{12} ML^2$
thin rod, length $L$	end, perpendicular	$\tfrac13 ML^2$
rectangular plate $a \times b$	center, perpendicular to plate	$\tfrac1{12} M(a^2 + b^2)$

Parallel-axis theorem: $I = I_\text{cm} + Md^2$ . Perpendicular-axis (flat plates): $I_z = I_x + I_y$ .

Gravitation

Quantity	Formula
Newton's law	$F = \dfrac{GMm}{r^2}$ , attractive
field strength	$g = \dfrac{GM}{r^2}$ ( $9.81\ \text{m/s}^2$ at Earth's surface)
potential energy	$U = -\dfrac{GMm}{r}$ (zero at infinity)
circular orbit speed	$v = \sqrt{GM/r}$
escape speed	$v_\text{esc} = \sqrt{2GM/r}$ ( $11.2\ \text{km/s}$ from Earth)
Kepler's third law	$T^2 = \dfrac{4\pi^2}{GM}r^3$ ( $r$ = semi-major axis for ellipses)
orbital energy	$E = -\dfrac{GMm}{2r}$

Kepler: orbits are ellipses with the Sun at a focus; the radius sweeps equal areas in equal times (angular momentum is conserved); $T^2 \propto a^3$ .

Oscillations and waves

Simple harmonic motion

Simple harmonic motion (SHM): restoring force proportional to displacement, $\ddot x = -\omega^2 x$ .

x(t) = A\cos(\omega t + \phi), \quad v(t) = -A\omega\sin(\omega t + \phi), \quad a(t) = -\omega^2 x

Quantity	Formula
angular frequency	$\omega = 2\pi f = 2\pi/T$
mass on spring	$\omega = \sqrt{k/m}$ , $\ T = 2\pi\sqrt{m/k}$
simple pendulum (small angle)	$\omega = \sqrt{g/L}$ , $\ T = 2\pi\sqrt{L/g}$
physical pendulum	$T = 2\pi\sqrt{I/(mgd)}$ , $d$ = pivot to center of mass
maxima	$v_\text{max} = A\omega$ , $\ a_\text{max} = A\omega^2$
energy	$E = \tfrac12 kA^2 = \tfrac12 kx^2 + \tfrac12 mv^2$
speed at $x$	$v = \omega\sqrt{A^2 - x^2}$
damped (light)	$x = Ae^{-bt/2m}\cos(\omega' t)$ , $\ \omega' = \sqrt{\omega^2 - (b/2m)^2}$

With $A = \omega = 1$ and $\phi = 0$ : velocity leads displacement by a quarter cycle and acceleration is always opposite to displacement.

SHM: x, v and a (A = ω = 1) x = cos t v = −sin t a = −cos t

Driven oscillator: resonance when the driving frequency is near $\omega_0$ ; lighter damping gives a taller, narrower peak.

Waves and sound

Quantity	Formula
wave speed	$v = f\lambda = \lambda/T$
traveling wave	$y = A\sin(kx - \omega t)$ , $\ k = 2\pi/\lambda$ , $\ v = \omega/k$
string	$v = \sqrt{F_T/\mu}$ , $\mu$ = mass per length
sound in air	$v \approx 331 + 0.6\,T_{^\circ\text{C}}$ , about $343\ \text{m/s}$ at 20 °C
intensity	$I = P/(4\pi r^2)$ from a point source
sound level	$\beta = 10\log_{10}(I/I_0)$ dB, $\ I_0 = 10^{-12}\ \text{W/m}^2$
beats	$f_\text{beat} = \lvert f_1 - f_2 \rvert$

Doppler effect (sound)

f' = f\,\frac{v \pm v_o}{v \mp v_s}

Top signs when observer and source move toward each other: the observed frequency rises. $v$ is the speed of sound in the medium. Light, for $v \ll c$ : $\Delta f / f \approx v_r / c$ .

Superposition and standing waves

Situation	Condition or modes
constructive interference	path difference $\Delta = m\lambda$
destructive interference	$\Delta = (m + \tfrac12)\lambda$
double slit, bright fringes	$d\sin\theta = m\lambda$ , spacing $\Delta y = \lambda L/d$
single slit, first dark fringe	$a\sin\theta = \lambda$
string fixed both ends, open pipe	$\lambda_n = 2L/n$ , $\ f_n = nv/2L$ , $n = 1, 2, 3, \ldots$
pipe closed one end	$\lambda_n = 4L/n$ , $\ f_n = nv/4L$ , $n = 1, 3, 5, \ldots$

Fluids

Quantity	Formula
density, pressure	$\rho = m/V$ , $\ p = F/A$ (1 atm $= 101.325$ kPa)
hydrostatic pressure	$p = p_0 + \rho g h$ ; gauge pressure is $p - p_\text{atm}$
Pascal (hydraulic press)	$F_1/A_1 = F_2/A_2$
buoyancy (Archimedes)	$F_b = \rho_\text{fluid} V_\text{sub} g$
floating fraction	$V_\text{sub}/V = \rho_\text{object}/\rho_\text{fluid}$
continuity	$A_1 v_1 = A_2 v_2$ (volume flow $Q = Av$ )
Bernoulli (along a streamline)	$p + \tfrac12\rho v^2 + \rho g h = \text{const}$
Torricelli	$v = \sqrt{2gh}$ out of a hole at depth $h$
Poiseuille (laminar pipe)	$Q = \dfrac{\pi r^4 \Delta p}{8\eta L}$
Reynolds number	$Re = \rho v D/\eta$ ; pipes are laminar below about 2000

Bernoulli assumes steady, incompressible, non-viscous flow. The atmosphere is the compressible, rotating case: see numerical weather modeling.

Thermodynamics

Quantity	Formula
temperature	$T_\text{K} = T_{^\circ\text{C}} + 273.15$
linear expansion	$\Delta L = \alpha L\,\Delta T$
heat to change temperature	$Q = mc\,\Delta T$ ( $c_\text{water} \approx 4186$ J/(kg·K))
latent heat (phase change)	$Q = mL$
conduction	$P = kA\,\Delta T / d$
radiation	$P = \varepsilon\sigma A T^4$
ideal gas	$pV = nRT = Nk_BT$
mean kinetic energy	$\langle K \rangle = \tfrac32 k_B T$ , $\ v_\text{rms} = \sqrt{3k_BT/m}$
internal energy, monatomic	$U = \tfrac32 nRT$

Laws

Law	Statement
zeroth	two bodies each in equilibrium with a third are in equilibrium with each other
first	$\Delta U = Q - W$ ( $Q$ into the system, $W$ done by it)
second	total entropy of an isolated system never decreases; heat flows hot to cold
third	entropy approaches a minimum as $T \to 0$ ; absolute zero is unreachable

Ideal gas processes

Process	Constant	Work by gas $W$	Note
isobaric	$p$	$p\,\Delta V$	$Q = nC_p\Delta T$
isochoric	$V$	$0$	$Q = \Delta U = nC_V\Delta T$
isothermal	$T$	$nRT\ln(V_2/V_1)$	$\Delta U = 0$ , $Q = W$
adiabatic	$pV^\gamma$	$\dfrac{p_1V_1 - p_2V_2}{\gamma - 1}$	$Q = 0$ , $\gamma = C_p/C_V$

$C_p - C_V = R$ ; monatomic $\gamma = 5/3$ , diatomic (air) $\gamma \approx 7/5$ .

Entropy and heat engines

Quantity	Formula
entropy change (reversible)	$\Delta S = \int dQ/T$ ; at constant $T$ : $\Delta S = Q/T$
engine efficiency	$\eta = W/Q_h = 1 - Q_c/Q_h$
Carnot (maximum) efficiency	$\eta_C = 1 - T_c/T_h$ (kelvin)
refrigerator COP	$K = Q_c/W$ , $\ K_C = T_c/(T_h - T_c)$
heat pump COP	$Q_h/W$ , $\ T_h/(T_h - T_c)$

Electricity and magnetism

Electrostatics

Quantity	Formula	Unit
Coulomb's law	$F = k\dfrac{q_1q_2}{r^2}$ , $\ k = \dfrac{1}{4\pi\varepsilon_0}$	N
field	$\vec E = \vec F/q$ ; point charge $E = kq/r^2$	N/C = V/m
potential	$V = kq/r$ ; $\ \Delta U = q\Delta V$	V
field from potential	$E_x = -\dfrac{dV}{dx}$ ; parallel plates $E = V/d$	V/m
Gauss's law	$\oint \vec E \cdot d\vec A = Q_\text{enc}/\varepsilon_0$
capacitance	$C = Q/V$ ; parallel plates $C = \varepsilon_0 \kappa A/d$	F
energy in a capacitor	$U = \tfrac12 CV^2 = \dfrac{Q^2}{2C}$	J

Energy unit: $1\ \text{eV} = 1.602 \times 10^{-19}\ \text{J}$ , the energy an electron gains across 1 V.

Circuits

Quantity	Formula
current	$I = dQ/dt$ (A)
Ohm's law	$V = IR$
resistance of a wire	$R = \rho L/A$
power	$P = IV = I^2R = V^2/R$
resistors in series	$R = R_1 + R_2 + \cdots$ (same current)
resistors in parallel	$\dfrac1R = \dfrac1{R_1} + \dfrac1{R_2} + \cdots$ (same voltage)
capacitors in series	$\dfrac1C = \dfrac1{C_1} + \dfrac1{C_2} + \cdots$
capacitors in parallel	$C = C_1 + C_2 + \cdots$
real battery	$V = \mathcal{E} - Ir$
AC RMS	$V_\text{rms} = V_0/\sqrt2$ , $\ I_\text{rms} = I_0/\sqrt2$

Kirchhoff: currents into a junction sum to zero (charge); voltage changes around any closed loop sum to zero (energy).

RC circuit, time constant $\tau = RC$ (63% of the way after one $\tau$ , over 99% after $5\tau$ ):

\text{charging: } V_C = V_0\left(1 - e^{-t/RC}\right), \qquad \text{discharging: } V_C = V_0\,e^{-t/RC}, \qquad I = \frac{V_0}{R}e^{-t/RC}

Magnetism and induction

Quantity	Formula
Lorentz force	$\vec F = q(\vec E + \vec v \times \vec B)$
force on a wire	$\vec F = I\vec L \times \vec B$
radius in a uniform field	$r = \dfrac{mv}{qB}$ , $\ f = \dfrac{qB}{2\pi m}$ (cyclotron)
long straight wire	$B = \dfrac{\mu_0 I}{2\pi r}$ (circles, right-hand grip)
solenoid	$B = \mu_0 n I$ , $n$ = turns per meter
flux	$\Phi_B = \vec B \cdot \vec A = BA\cos\theta$ (Wb)
Faraday + Lenz	$\mathcal{E} = -N\dfrac{d\Phi_B}{dt}$ ; induced current opposes the change
motional emf	$\mathcal{E} = BLv$
generator	$\mathcal{E} = NBA\omega\sin\omega t$
transformer (ideal)	$V_s/V_p = N_s/N_p = I_p/I_s$
inductor	$\mathcal{E} = -L\,dI/dt$ , energy $\tfrac12 LI^2$ , RL $\tau = L/R$

Maxwell's equations tie all of this together and predict light at speed $c = 1/\sqrt{\mu_0\varepsilon_0}$ .

Optics

Quantity	Formula
reflection	$\theta_i = \theta_r$ (from the normal)
refractive index	$n = c/v$ ; in a medium $\lambda_n = \lambda/n$ , $f$ unchanged
Snell's law	$n_1\sin\theta_1 = n_2\sin\theta_2$
critical angle	$\sin\theta_c = n_2/n_1$ (needs $n_1 > n_2$ ; beyond it: total internal reflection)
thin lens and mirror	$\dfrac1f = \dfrac1{d_o} + \dfrac1{d_i}$
magnification	$m = -\dfrac{d_i}{d_o} = \dfrac{h_i}{h_o}$
spherical mirror	$f = R/2$
lensmaker	$\dfrac1f = (n - 1)\left(\dfrac1{R_1} - \dfrac1{R_2}\right)$
lens power	$P = 1/f$ in dioptres ( $f$ in meters)
diffraction grating	$d\sin\theta = m\lambda$

Sign convention (real is positive)	Positive	Negative
$f$	converging lens, concave mirror	diverging lens, convex mirror
$d_i$	real image	virtual image
$m$	upright	inverted

Modern physics

Pointers only; each line opens a whole subject.

Idea	Formula
mass-energy	$E = mc^2$ ; $\ E^2 = (pc)^2 + (mc^2)^2$
Lorentz factor	$\gamma = 1/\sqrt{1 - v^2/c^2}$
time dilation, length contraction	$\Delta t = \gamma\Delta t_0$ , $\ L = L_0/\gamma$
photon energy	$E = hf = hc/\lambda$ ; $hc \approx 1240\ \text{eV·nm}$
photon momentum	$p = E/c = h/\lambda$
photoelectric effect	$K_\text{max} = hf - \phi$
de Broglie wavelength	$\lambda = h/p = h/mv$
uncertainty	$\Delta x\,\Delta p \ge \hbar/2$
hydrogen levels	$E_n = -13.6\ \text{eV}/n^2$
Wien's law	$\lambda_\text{max}T = 2.898 \times 10^{-3}\ \text{m·K}$
radioactive decay	$N = N_0e^{-\lambda t}$ , $\ t_{1/2} = \ln 2/\lambda$

Worked examples

Projectile range

A ball leaves level ground at $v_0 = 20$ m/s, $\theta = 30^\circ$ . Find flight time, range and peak height.

\begin{aligned} v_x &= 20\cos 30^\circ = 17.32\ \text{m/s}, \qquad v_y = 20\sin 30^\circ = 10.0\ \text{m/s} \\ T &= \frac{2v_y}{g} = \frac{2(10.0)}{9.81} = 2.04\ \text{s} \\ R &= v_xT = 17.32 \times 2.039 = 35.3\ \text{m} \quad \left(= \tfrac{v_0^2\sin 60^\circ}{g}\right) \\ H &= \frac{v_y^2}{2g} = \frac{100}{19.62} = 5.10\ \text{m} \end{aligned}

Block sliding down a rough incline

$m = 5.0$ kg on a $30^\circ$ slope, $\mu_k = 0.20$ . Axes: $x$ down the slope.

\begin{aligned} N &= mg\cos\theta = 5.0(9.81)(0.866) = 42.5\ \text{N} \\ f_k &= \mu_k N = 0.20(42.5) = 8.50\ \text{N} \\ ma &= mg\sin\theta - f_k = 24.5 - 8.50 = 16.0\ \text{N} \\ a &= 16.0/5.0 = 3.21\ \text{m/s}^2 \end{aligned}

Car power from kinetic energy

A 1200 kg car reaches 100 km/h from rest in 8.0 s. Average power, ignoring losses:

v = \frac{100}{3.6} = 27.8\ \text{m/s}, \quad K = \tfrac12(1200)(27.8)^2 = 4.63 \times 10^5\ \text{J}, \quad P = \frac{K}{t} = 57.9\ \text{kW}

Elastic and sticky collisions

$m_1 = 2.0$ kg at $3.0$ m/s hits $m_2 = 1.0$ kg at rest.

Case	After	$K$ before → after
perfectly inelastic	$v' = \dfrac{2.0(3.0)}{3.0} = 2.0$ m/s	9.0 J → 6.0 J
elastic	$v_1' = \tfrac{1}{3}(3.0) = 1.0$ m/s, $\ v_2' = \tfrac{4}{3}(3.0) = 4.0$ m/s	9.0 J → 9.0 J

Check momentum: $2.0(1.0) + 1.0(4.0) = 6.0\ \text{kg·m/s}$ both times.

Geostationary orbit radius

Period one sidereal day, $T = 86\,164$ s; $GM_\oplus = 3.986 \times 10^{14}\ \text{m}^3/\text{s}^2$ .

r = \left(\frac{GM T^2}{4\pi^2}\right)^{1/3} = \left(\frac{3.986 \times 10^{14}\,(86\,164)^2}{39.48}\right)^{1/3} = 4.22 \times 10^7\ \text{m}

Altitude above the equator: $42\,164 - 6378 \approx 35\,786$ km. Orbital speed $2\pi r/T = 3.07$ km/s.

RC charging

$R = 10\ \text{k}\Omega$ , $C = 100\ \mu\text{F}$ , supply $5.0$ V.

\tau = RC = (10^4)(10^{-4}) = 1.0\ \text{s}, \qquad V_C(2\ \text{s}) = 5.0\left(1 - e^{-2}\right) = 4.32\ \text{V}

Time to reach 99%: $1 - e^{-t/\tau} = 0.99 \Rightarrow t = \tau\ln 100 = 4.6$ s.

Thin lens image

Converging lens $f = 10$ cm, object at $d_o = 15$ cm.

\frac{1}{d_i} = \frac{1}{10} - \frac{1}{15} = \frac{1}{30} \ \Rightarrow\ d_i = 30\ \text{cm}, \qquad m = -\frac{30}{15} = -2

Real, inverted, twice the size, on the far side of the lens.

References

OpenStax: University Physics Volume 1 (opens in a new tab): mechanics, waves, oscillations; source of the sign conventions used here
OpenStax: University Physics Volume 2 (opens in a new tab): thermodynamics, electricity and magnetism
OpenStax: University Physics Volume 3 (opens in a new tab): optics and modern physics
HyperPhysics (opens in a new tab): concept map with a calculator on most formula pages
The Feynman Lectures on Physics (opens in a new tab): the reasoning behind the formulas, free online
NIST: CODATA fundamental physical constants (opens in a new tab): authoritative values for $G$ , $h$ , $e$ , $k_B$
NIST SP 330: The International System of Units (opens in a new tab): SI base and derived units
Constants, units and conversions and notation: companion sheets