Physics 2AG Formula Sheet 1 Kinematics in 1D, 2D Kinematics Equations v = v₀ + at v² = v₀² + 2a(x - x₀) x = x₀ + v₀t + ½at² x = x₀ + ½(v₀ + v)t Free Fall v_y = v_{0y} - gt v_y² = v_{0y}² - 2g(y - y₀) y = y₀ + v_{0y}t - ½gt² R = (v₀²/g)sin(2θ) Projectile Motion x = (v₀cosθ)t v_x = v₀sinθ - gt v_x² = (v₀sinθ)² - 2gΔy y = y₀ + (v₀sinθ)t - ½gt² 2 Forces & Newton's Laws Newton's 2nd Law ΣF⃗ = ma⃗ or ΣF = 0 ΣF_x = ma_x or ΣF_x = 0 ΣF_y = ma_y or ΣF_y = 0 Friction f_s ≤ μ_sN f_k = μ_kN Gravitational Force F_G = G(m₁m₂)/r² g_{planet} = G(M/R²) W_{Apparent} = mg + ma 3 Uniform Circular Motion Centripetal Force and Acceleration a_c = v²/r = (2πr/T)²/r = (2πr f)²/r T = 2πr/v v = 2πrf a_c = v²/r F_c = ma_c = m(v²/r) Banked Curves tanθ = v²/rg v = √(rgtanθ) Satellites in Circular Orbits v = √(GM_E/r) T = (2πr³/²)/√(GM_E) 4 Work & Energy Work W = (Fcosθ)s W = W_F - K_0 Power P = Work/Time P = ΔE/Δt P = Fv Kinetic Energy K = ½mv² Work-Energy Theorem W_{net} = ΔK W_{nc} = ΔE U_G = mgh U_S = ½kx² E = U + K W_{net} = ΔK F_s = -k_x 5 Impulse & Momentum Impulse-Momentum Theorem p⃗ = mv⃗ I⃗ = F⃗Δt = Δp⃗ F⃗_netΔt = Δp⃗ F⃗_net = mv⃗_f - mv⃗_0 Conservation on Linear Momentum p⃗ = Σp⃗_i = Σp⃗_f m₁v⃗_{01} + m₂v⃗_{02} = m₁v⃗_{f1} + m₂v⃗_{f2} (2D with subscripts x, y) 6 Rotational Kinematics θ = s/r ω̄ = Δθ/Δt ᾱ = Δω/Δt Elastic Collisions ½m₁v_{01}² + ½m₂v_{02}² = ½m₁v_{f1}² + ½m₂v_{f2}² Elastic Head-On Col. v_{f1} = ((m₁-m₂)/(m₁+m₂))v_{01} + ((2m₂)/(m₁+m₂))v_{02} v_{f2} = ((2m₁)/(m₁+m₂))v_{01} + ((m₂-m₁)/(m₁+m₂))v_{02} Perfectly Inelastic Col. m₁v_{01} + m₂v_{02} = (m₁ + m₂)v_f Center of Mass X⃗_{CM} = (Σm_i x⃗_i)/(Σm_i) V⃗_{CM} = (Σm_i v⃗_i)/(Σm_i) A⃗_{CM} = (Σm_i a⃗_i)/(Σm_i) p⃗ = MV⃗_{CM} ΣF⃗_{ext} = MA⃗_{CM} Angular/Tangential Variables s = rθ v_T = rω a_T = rα a_c = rω² = v_T²/r Rolling v = v_T = rω a = a_T = rα Energy E = ½mv² + ½Iω² + mgh + ½kx² 7 Rotational Dynamics Center of Gravity X_{CG} = (W₁x₁ + W₂x₂ + ...)/(W₁ + W₂ + ...) Newton's 2nd - Rotation τ = Flsinθ τ = Στ = m₁r₁²α + m₂r₂²α + ... Στ = Iα Work & Energy W_R = τθ K_R = ½Iω² E = ½mv² + ½Iω² + mgh Angular Momentum L = Iω Στ = Iα = ΔL/Δt L_i = L_f I_iω_i = I_fω_f Work and Power W = τΔθ W = ΔK 8 Gravitation F_G = (Gm₁m₂)/R² g_{planet} = (GM_{planet})/R_{planet}² v_{esc} = √(2GM_{planet}/R_{planet}) Kepler T = (2π/√(GM))r³/² Energy U = -(Gm₁m₂)/r E = K + U = ½mv² - (Gm₁m₂)/r E = ½mv² + ½Iω² - (Gm₁m₂)/r + mgh + ½kx² 9 Simple Harmonic Motion (SHM) f = 1/T ω = 2πf = 2π/T x = Acos(ωt + φ) v = -ωAsin(ωt + φ) a = -ω²Acos(ωt + φ) v_{max} = Aω a_{max} = Aω² Spring F = -kx T = 2π√(m/k) U = ½kx² E = ½mv² + ½kx² + mgh + ½Iω² 10 Waves and Sound v = λf v = λ/T v_{string} = √(Tension/μ) Standing Waves λ_n = 2L/n (string) λ_n = 4L/(2n-1) (tube, 1 end closed) λ_n = 2L/n (tube, open ends) n = 1, 2, 3... 11 Fluids ρ = m/V Pendulum T = 2π√(L/g) (simple) T = 2π√(I/mgL_{cm}) (physical) Pascal's Principle P = F/A F₁/A₁ = F₂/A₂ A₁d₁ = A₂d₂ P = F₁/A₁ = F₂/A₂ P_B = P₂ + ρgh F_B = ρV_f g; ρV_f g d_f/d_s Continuity and Bernoulli P₁A₁v₁ = P₂A₂v₂ P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂ 12 Temperature and Heat ΔL/L₀ = αΔT ΔA/A₀ = 2αΔT ΔV/V₀ = 3αΔT = βΔT C = Q/ΔT c = Q/mΔT Q_f = ±mL_f Q_v = ±mL_v T_C = (T_F - 32) (5/9) T_F = (9/5)T_C + 32°F Q_{Cond} = kA(ΔT/L)t P_{Rad} = σεAT⁴ P_{net} = σεA(T⁴ - T_s⁴) 13 Constants & Conversions C = 2πr A_{circle} = πr² A_{sphere} = 4πr² V_{sphere} = (4/3)πr³ V_{cylinder} = πr²h g = 9.8 m/s² 1 mi = 1.609 km 1 ft = 0.305 m 1 kg = 2.2 lb G = 6.674 x 10⁻¹¹ N·m²/kg² 1 atm = 1.013 x 10⁵ Pa 1 bar = 10⁵ Pa 1 Pa = 7.5 x 10⁻³ mmHg v_{sound} = 343 m/s 1 cal = 4.186 J 1 kcal = 1 C = 4186 J T_K = T_C + 273.15 37.7s