MTEL Physics (69) cheat sheet

Constant-acceleration kinematics: v = v₀ + at, Δx = v₀t + ½at², and v² = v₀² + 2aΔx; all three fail once acceleration is not constant.

Newton's second law ΣF = ma: acceleration is parallel to the net force and inversely proportional to mass; a = 0 whenever the net force is zero, even if several individual forces act.

Projectile motion: with air resistance neglected, horizontal velocity stays constant while vertical motion has a = −g ≈ −9.8 m/s²; the components are independent and recombine as vectors.

Momentum p = mv and impulse J = FΔt = Δp; total momentum is conserved when the net external force on the system is zero.

Elastic collisions conserve momentum and kinetic energy; perfectly inelastic collisions conserve momentum only and leave the bodies moving together at v = (m₁v₁ + m₂v₂)/(m₁ + m₂).

Uniform circular motion: a_c = v²/r directed toward the center; the centripetal force is the net inward force (tension, gravity, friction), never a separate extra force.

Rotational analogs: τ = Iα, L = Iω, and K_rot = ½Iω²; the moment of inertia depends on both the mass distribution and the chosen axis.

Hooke's law F = −kx gives SHM with T = 2π√(m/k), independent of amplitude; a simple pendulum has T = 2π√(L/g) for small angles, independent of mass.

Fluids: ρ = m/V; continuity A₁v₁ = A₂v₂; Bernoulli P + ½ρv² + ρgh is constant along a streamline; buoyant force F_b = ρ_fluid·V_submerged·g.

Wave relation v = fλ; on a stretched string v = √(F_T/μ), so raising tension or using a lighter string increases speed while frequency stays source-determined.

Standing waves: string or open pipe f_n = nv/(2L) for n = 1, 2, 3…; a pipe closed at one end gives f_n = nv/(4L) for odd n only.

Doppler effect f′ = f(v ± v_o)/(v ∓ v_s); observed frequency rises when source and observer approach and drops when they separate.

Refraction: n = c/v and Snell's law n₁ sin θ₁ = n₂ sin θ₂; light entering a higher-index medium bends toward the normal, and total internal reflection occurs only from high to low index when θ₁ > θ_c with sin θ_c = n₂/n₁.

Thin lenses and mirrors: 1/f = 1/d_o + 1/d_i with m = −d_i/d_o; a positive image distance means a real image (opposite side for lenses, same side for mirrors).

Interference: double-slit maxima at d sin θ = mλ; single-slit minima at a sin θ = mλ (m ≠ 0); longer wavelength or smaller slit spacing spreads the pattern.

Ideal gas PV = nRT with R = 8.31 J/(mol·K); average translational kinetic energy per molecule is (3/2)k_BT, depending only on temperature.

First law ΔU = Q − W with W done by the gas: adiabatic Q = 0, isothermal ΔU = 0 for an ideal gas, isochoric W = 0; heating uses Q = mcΔT and phase change uses Q = mL.

Second law and efficiency: entropy of an isolated system never decreases; e = W/Q_H = 1 − Q_C/Q_H, and the Carnot limit is e_max = 1 − T_C/T_H with kelvin temperatures.

Coulomb's law F = k|q₁q₂|/r² with k ≈ 8.99 × 10⁹ N·m²/C²; the force is repulsive for like charges and falls off as the inverse square of separation.

Point-charge field E = kq/r² points radially away from positive charge and toward negative charge; F = qE gives the force on a test charge and field lines never cross.

Potential V = kq/r is a scalar; W = qΔV, equipotentials are perpendicular to field lines, and E always points toward decreasing potential.

Capacitance C = Q/V; parallel capacitors add directly, series capacitances add as reciprocals, and stored energy is U = ½CV² = Q²/(2C).

Resistors: series R_eq = ΣR, parallel 1/R_eq = Σ1/R; Ohm's law V = IR and power P = IV = I²R = V²/R.

Kirchhoff's rules: the junction rule expresses charge conservation and the loop rule expresses energy conservation; use them when a circuit cannot be reduced by series/parallel combinations.

Magnetic force F = qvB sin θ or F = ILB sin θ is perpendicular to both the velocity (or current) and B; the right-hand rule gives direction, and a charge moving along B feels no force.

Field of a long straight wire B = μ₀I/(2πr); inside a solenoid B = μ₀nI, where n is turns per meter.

Faraday's law ε = −N ΔΦ/Δt; Lenz's law makes the induced current oppose the flux change, and a rod of length ℓ moving at v has motional emf ε = Bℓv.

Photon energy E = hf = hc/λ with h = 6.63 × 10⁻³⁴ J·s and hc ≈ 1240 eV·nm; visible photons carry roughly 1.8–3.1 eV.

Photoelectric effect: K_max = hf − φ; no electrons appear below the threshold f₀ = φ/h regardless of intensity, and the stopping potential satisfies eV_s = K_max.

de Broglie wavelength λ = h/p; electrons diffract at atomic spacings, which is why electron microscopes resolve far better than optical ones.

Bohr model for hydrogen: E_n = −13.6 eV/n²; a photon is emitted when an electron drops to a lower level, with energy equal to the level difference.

Mass-energy E = mc²; a nuclear mass defect appears as binding energy, and 1 atomic mass unit equals 931.5 MeV.

Radioactive decay N = N₀e^(−λt); half-life T½ = ln2/λ ≈ 0.693/λ, and activity A = λN is measured in becquerels (1 Bq = 1 decay per second).

Decay modes: alpha emission lowers A by 4 and Z by 2; beta-minus raises Z by 1 with A unchanged; gamma emission changes neither A nor Z.

Nuclear reactions must balance charge, nucleon number (A), baryon number, lepton number, and energy-momentum.

Relativity: γ = 1/√(1 − v²/c²); moving clocks dilate (Δt = γΔt₀) and lengths contract along the motion (L = L₀/γ), and no massive object reaches c.