Praxis Physics 5265 cheat sheet

Newton's second law: ΣF = ma; for constant acceleration, use v = v₀ + at, Δx = v₀t + ½at², v² = v₀² + 2aΔx.

Projectile motion: horizontal velocity is constant (a_x = 0); vertical acceleration is g downward; time of flight depends only on vertical motion.

Circular motion: centripetal acceleration a_c = v²/r directed toward the center; centripetal force is the net inward force, not a separate force.

Work-energy theorem: W_net = ΔK = ½mv² − ½mv₀²; for conservative forces, mechanical energy E = K + U is conserved.

Momentum: p = mv; impulse J = FΔt = Δp; in an isolated system total momentum is conserved in all collisions.

Elastic collision: kinetic energy and momentum both conserved; perfectly inelastic collision: objects stick together, only momentum conserved.

Torque τ = rF sinθ; static equilibrium requires ΣF = 0 and Στ = 0 about any pivot.

Universal gravitation: F = Gm₁m₂/r²; orbital speed for circular orbit v = √(GM/r); Kepler's third law T² ∝ r³.

Coulomb's law: F = kq₁q₂/r², k ≈ 8.99×10⁹ N·m²/C²; electric field E = F/q₀, with field lines pointing away from positive charge.

Electric potential energy U = kq₁q₂/r; potential V = kq/r; work done by external agent W = qΔV.

Gauss's law: Φ_E = ∮E·dA = Q_enclosed/ε₀; useful for spherical, cylindrical, and planar symmetry.

Capacitance C = Q/V; parallel-plate C = ε₀A/d; energy stored U = ½CV² = Q²/(2C).

Ohm's law V = IR; resistors in series add (R_eq = R₁ + R₂ + …); in parallel, 1/R_eq = 1/R₁ + 1/R₂ + ….

Kirchhoff's rules: junction rule ΣI_in = ΣI_out; loop rule ΣΔV = 0 around any closed loop.

Magnetic force on moving charge F = qv × B (magnitude qvB sinθ); on current-carrying wire F = IL × B.

Faraday's law: induced emf ε = −N ΔΦ_B/Δt; Lenz's law: induced current opposes the change in magnetic flux.

Wave relation v = fλ; for a wave on a string v = √(T/μ), where T is tension and μ is linear mass density.

Standing waves on a string fixed at both ends: λ_n = 2L/n, f_n = nv/(2L), n = 1, 2, 3, ….

Sound: speed in air ≈ 343 m/s at 20 °C; open pipe has antinodes at both ends; closed pipe has a node at the closed end and antinode at the open end.

Doppler effect for sound: f' = f (v ± v_o)/(v ∓ v_s), with signs chosen so motion toward the observer raises the observed frequency.

Snell's law: n₁ sinθ₁ = n₂ sinθ₂; total internal reflection occurs when light travels from higher to lower n and θ₁ exceeds the critical angle θ_c = sin⁻¹(n₂/n₁).

Thin lens/mirror equation: 1/f = 1/d_o + 1/d_i; magnification m = −d_i/d_o; real images form on the same side as the outgoing rays for mirrors and opposite side for lenses.

Young's double slit: bright fringes at d sinθ = mλ; single-slit minima at a sinθ = mλ (m ≠ 0).

Diffraction grating: maxima at d sinθ = mλ; resolving power R = λ/Δλ = mN.

Ideal gas law: PV = nRT = Nk_BT; R ≈ 8.31 J/(mol·K), k_B ≈ 1.38×10⁻²³ J/K.

Kinetic theory: average translational kinetic energy per molecule = (3/2)k_BT; root-mean-square speed v_rms = √(3RT/M).

First law of thermodynamics: ΔU = Q − W, where W is work done by the system; for a monatomic ideal gas U = (3/2)nRT.

Thermodynamic processes: isothermal (ΔT = 0, ΔU = 0), adiabatic (Q = 0), isobaric (W = PΔV), isochoric (W = 0).

Heat transfer: Q = mcΔT for temperature change; Q = mL for phase change at constant temperature.

Carnot efficiency: e = 1 − T_c/T_h (temperatures in kelvin); no real engine between the same reservoirs can exceed this.

Entropy change for reversible process: ΔS = Q_rev/T; second law: ΔS_universe ≥ 0.

Thermal expansion: ΔL = αL₀ΔT (linear); ΔV = βV₀ΔT with β ≈ 3α for solids.

Photon energy E = hf = hc/λ; h ≈ 6.63×10⁻³⁴ J·s; hc ≈ 1240 eV·nm.

Photoelectric effect: K_max = hf − φ, where φ is the work function; no emission below threshold frequency f₀ = φ/h.

de Broglie wavelength λ = h/p = h/(mv); for an electron accelerated through potential V, λ ≈ 1.23 nm/√(V in volts).

Bohr model: E_n = −13.6 eV/n² for hydrogen; photon energy for transition = |E_f − E_i|.

Mass-energy equivalence E = mc²; nuclear binding energy is the mass defect times c².

Radioactive decay: N = N₀e^(−λt); half-life t₁/₂ = ln2/λ ≈ 0.693/λ; activity A = λN.

Heisenberg uncertainty principle: ΔxΔp ≥ ħ/2, where ħ = h/(2π).

Special relativity: γ = 1/√(1 − v²/c²); time dilation Δt = γΔt₀; length contraction L = L₀/γ.

Significant figures: for multiplication/division, result has the same number of sig figs as the least precise factor; for addition/subtraction, same number of decimal places.

Uncertainty propagation: for sums/differences add absolute uncertainties; for products/quotients add fractional uncertainties; for powers multiply fractional uncertainty by the exponent.

Precision describes repeatability of measurements; accuracy describes closeness to the accepted value; systematic error shifts all readings in one direction.

Graph linearization: plot variables so the expected relation is y = mx + b; slope and intercept then yield physical constants with units.

Percent error = |experimental − accepted|/accepted × 100%; percent difference compares two experimental values.

A controlled experiment changes one independent variable at a time while holding others constant; a control group receives no treatment.

Laboratory safety: know location of eyewash, fire extinguisher, and safety shower; never taste or directly smell chemicals; report spills immediately.

Calibration checks a measuring instrument against a known standard; a zero reading (tare) must be recorded and subtracted when using a digital balance.