Equations

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Kinematic Equations
Gravitational Equations
Physical Equations
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Thermodynamic Equations
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Atomic Equations
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Special Relativistic Equations (not finished)
General Relativistic Equations (not finished)
Cosmological Equations (not finished)
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Mathematical Equations (not finished)
Dimensional Constant Equations (not finished)
Dimensionless Constant Equations (not finished)

a = Δv/Δt = F/m = (v_f − v_i)/Δt = w/m

a_c = 4π²r/P² = rω² = v²/r

ad = v²/2
aΔt = Δv = v_f − v_i
a = Δv/Δt = (v_f − v_i)/Δt
α = a/r = Δω/Δt = T/I (T = torque)
αΔt = ω_f − ω_i
α = (ω_f − ω_i)/Δt
d = ½(v_i + v_f)Δt = (v_i + ½aΔt)Δt = v_iΔt + ½aΔt² = vΔt
d = ½gΔt² = vΔt = ((v_i + v_f)/2)Δt (free fall)
d = W/F
d_f = d_i + Δt((v_i + v_f)/2) = d_i + v_fΔt − ½aΔt² = d_i + v_iΔt + (aΔt²/2) (distance traveled after acceleration; linear equations of motion)
Δφ = l_arc/r = ω_iΔt + ½αΔt² = ωΔt
Δp = Ft = mv_f − mv_i = m(v_f − v_i)



Δt = 2vsin(θ)/g (to hit ground)
Δt = (v_f − v_i)/a
Δv = v_exln(m_i/m_f) = −v_exln(m/(m_fuel + m)) (Tsiolkovsky rocket equation)
Δx = v₀Δtcos(θ)
Δx_tot = (v₀²/g)sin(2θ)
Δy = v₀Δtsin(θ) − ½gΔt²
E_K = Iω²/2 = ½mv² + ½Iω² = ½mv_x² + ½mv_y² + ½mv_z² = Fd = mad = mv²/2 = wd
f = P⁻¹

φ_f = φ_i + ½(ω_i + ω_f)Δt = φ_i + ω_fΔt − ½αΔt² = φ_i + ω_iΔt + ½αΔt² (angular displacement; rotational equations of motion)
F = ma = m(v_f − v_i)/Δt
F² = F₁² + F₂² − 2F₁F₂cos(θ_F1F2)
F₁/sin(θ_F2F3) = F₂/sin(θ_F1F3) = F₃/sin(θ_F1F2)
F_c = −4π²mr/P² = −ma_c = −mrω² = −mv²/r
F_Cor = ma_Cor
h = vΔtsin(θ) − ½gt²
h_max = v²sin²(θ)/(2g)
j = a/Δt
L = Iω = rp
L² = m²v²r²
m = F/a = w/g
p = mv = (p_x² + p_y² + p_z²)^1/2
P = 1/f = 2π/ω = 2πr/v
P² = 4π²r/a_c
P_Foucault = P_rot/sin(φ)
t = p²
t_hmax = vsin(θ)/g
torque = Iα




v = 2πr/P = L/(mr) = ωr

v² = v_x² + v_y² + v_z²
v_0y = vsin(θ)
v_AB = v_A − v_B (relative)
v_ex = I/m
v_f = v_i + aΔt (final velocity after time Δt)
v_f² = v_i² + 2aΔd (Torricelli's equation; linear equations of motion)
v_f − v_i = aΔt
v_f² = v_i² + 2ad
v_f² − v_i² = 2ad = 2gd
v_f − v_i = aΔt
vP = 2πr
v(t) = v_exln(m(0)/m(t)) − gt
v_x = v_0x = v₀cos(θ)
v_y = v_0y − gΔt
v = Δd/Δt = v_i + ½aΔt = (v_i + (v_i + aΔt))/2 = (v_i + v_f)/2 (average velocity during acceleration)
w = ma = mg
ω = Δφ/Δt = 2πf = 2π/P = v/r

ω_f = ω_i + αΔt (final angular velocity after time Δt; rotational equations of motion)
ω_f − ω_i = αΔt
ω_f² = ω_i² + 2αΔφ (rotational equations of motion)
ω_f² − ω_i² = 2αΔφ
ω = ½(ω_i + ω_f) = Δφ/Δt
yank = mj = F/Δt
W = Fd
z(t) = z₀ + v₀(t − t₀) − ½g(t − t₀)²

E_BG = −Gm₁m₂/d
E_Ke = ½mv_e² = −E_BG = Gm₁m₂/d
E_M = ½mv² − Gm₁m₂/r = −Gm₁m₂/(2r)
E_PG = −Gm₁m₂/d = mgh = −wh
φ = −Gm(3r² − d²)/(2r³) (inside homogenous sphere)
φ = −Gm/d
F_G = −Gm₁m₂/d² = −mg = −mv²/r = −w (law of universal gravitation)
F_G/F_E = −Gm₁m₂/(K_eq₁q₂)
F_T/m₂ = (4Gm₁dr₁)/(d² − r₁²)² ~ 4Gm₁r₁/d³

g = F_G/m = −Gm/d² = −v²/r
m = 4π²r³/(GP²)
m = v_e²d/(2G) = −gr²/G = −w/g
μ = Gm


P² = (4π²/(Gm))r³ = 4π²r/a_c
r = (2Gm)^1/3P^2/3/(2π^2/3)


v_e² = 2Gm/d
w = Gm₁m₂/d² = ma = −mg
W = E_PG2 − E_PG1 = Gm₁m₂((1/d₂) − (1/d₁))

d = (M/(2ρN_A))^1/3 (between ions)
E = E_K + E_P + E_U
E_X = E_K + E_P
I = (2/5)mr² (sphere)
k₀ = ½mv² + PV + mgh (Bernoulli's equation)
k₀ = ½ρv² + P + ρgh (Bernoulli's equation)
m = MN
m = ρV
M = m/N = N_Am
p = (ρ₀ − ρ)/ρ₀ (ρ₀ = pore-free)
P = E/t
P = F/A
P_l = gρh = ρ_wh
ρ = m/V
t_cr = (6 × 10⁸ a) × [(r_cluster[Mpc])/(v[Mm/s])]
V = M/(2ρN_A) (ion)
V = m/ρ

d[pc] = 10(1/10)^M/510^m/5

f = f₀/[1 + (v/c)cos(θ)]
f = c/λ (EM waves) = v/λ
Φ_v = ∫(E_v)dA
I_v = I_v0cos²(θ) (transmission of polarized light through polarizer, θ = between light & polarizer)
λ = λ₀(1 ± (v/c))
λ = v/f
λ_n = 2(l/n)
L = L_Sol10^(M_Sol10^3/5/10)(1/10)^(M10^3/5/10)

m = (M − 5) + 5log(d/pc)
m = m₁ − 2.5log{1 + antilog[−0.4(m₂ − m₁)]}
M = −(5ln(d/pc) − (m + 5)ln(10))/ln(10) = 5 + m − 5log₁₀(d/pc) = M_Sol − (10^2/5ln(L/L_Sol)/ln(10))

M₁ − M₂ = 100^1/5log(L₂/L₁)
M_Sol − M = 100^1/5log(L/L_Sol)
n = c/c₀
r_E = (2/c)√(Gm_deflectord_sourcex(1 − x)) (x = d_deflector/d_source)
S_a = ρv_a²/V
v = fλ
v_a = 20.1√(T/°K) = √(1.4P/ρ) = √(1.4RT/m) (gas)

ΔE_U = ΔH − PΔV
E = (2/3)nE_K = kT ≈ Δp_γc
E_a = E_i(1 − A) (grey body)
E_A = E_U − TS
E_B = TS
E_D = E_X + E_Q
E_G = E_V + E_A = H − TS
E_K = ½mv² = (3/2)kT
E_Q = E_A + E_B
E_Y = E_X + E_A
η = (1/3)ρv_rmsλ
φ = λ⁻¹
Φ_e = 4πr²σT⁴ = eσT⁴A
Φ_e = ∫(E_e)dA
H = E_U + PV
H_e = E_et
κ = λ/(cρ)
K = (log₁₀(P/W) − 6)/10
λ = 1/(π√(2)ρ_nd²) (d = of mol.) = kη/(ρv_rms) (k = 1/3−½) = kT/(π√(2)Pd²)
λ = π²k²γT/(3e²)
λ_max = b/T
log(L/L_Sol) = 3.3log(m/m_Sol)
L = L_Solm³(m/m_Sol)^0.3/m_Sol³
m = m_Sol(L/L_Sol)^10/33
M = ρRT/P (ideal gas law)
M_e = σT⁴
M_e/λ = c₁λ⁻⁵e^{−c₂/(λT)} = c₁λ⁻⁵/(e^hc/(λkT) − 1)
n_v/n = (4/√(π))(m/(2kT))^3/2v²e^−mv2/(2kT) (n_v = n having speed v)
n_v/n = (8πm³/h³)(v²/(e^{(mv2/2) − E_max} + 1)) (Fermi distribution)
N = PV/(RT) (ideal gas law)
NRT = (P + (N²a/V²))(V − Nb) (van der Waals equation)
P = (2/3)(n/V)E_K = F/A = (n/3)(mv_rms²/V) = (NRT/(V − Nb)) − (N²a/V²) (van der Waals equation) = (NR/V)T (ideal gas law) = ρ_nkT
P_ideal = P + (N²a/V²)
P₁V₁/T₁ = P₂V₂/T₂ (ΔN = 0) (combined gas law)
PM = ρRT (ideal gas law)
PV = NRT (universal gas equation)
q = cmΔT = CΔT = C(T_f − T_i) = ΔH = ΔE_U + PΔV = H_f − H_i
ρ = PM/RT (ideal gas law)
ρ_E = (4/c)σT⁴
(R/N_A)T = (2/3)(½mv²)
S = (5/2) + ln[(2πm_ukT/h²)^3/2kT/P]
τ = (10¹⁰ a)(m/m_Sol)⁻³
T = b/λ_max
T = M_e^1/4/σ^1/4 = √(2)L^1/4/(2√(r)(πσ)^1/4)
Tλ/γ = (π²/3)(k/e)²
v = √(2kT/m)
v_rms = √(3kT/m)
v_rms² = 3kT/m
v = √(8kT/(πm))
V_ideal = V − Nb
V = NRT/P (ideal gas law)

a = qE/m = qV/(dm) (of q between 2 V's separated by d)
a = (q/m)(Bv)
a = (q/m)γ⁻¹[E + vB − (v/c)(vE/c)]
B = μ₀I/(2πd) (straight wire)
B = μ₀nI/(2r) (center of circular coil of n turns)
B = F_B/(qvsin(θ)) = μ₀H + J = μ₀(H − M) = μH = Φ/A = √(B_x² + B_y² + B_z²) = √(2B_xy² + B_z²)
B = X⁻¹
B_z = B_xytan(φ)
χ_e = ε_r − 1
χ_m = μ_r − 1 = M/H
C = 4πεr (sphere)
C = q/V
D = εE
ε = D/E = ε₀ε_r
ε_r = ε/ε₀
E = ½CV² = ½(q²/C) = ½Vq (E stored in capacitor)
E = ½LI² = ∫(EI)dt (E of I stored in B)
E = D/ε = F_E/q = q/(4πε₀r²) = σ/ε
E_K = mv²/2 = (qBr)²/(2m) = qV
E_PE = ½CV² = ½q²/C = ½qV (capacitor)
E_PE = K_eq₁q₂/r = mv² = qV
E_T = I²Rt
f = Bq/(2πm)
F = g(B − Ev)
F_B = Bqv = Bqvsin(θ_Bv) = Gg/(4πμd²) = mv²/r
F_E = Eq = K_eq₁q₂/r² = mv²/r = q₁q₂/(4πεd²) = qVB
F_EM = q(E + Bv)
Φ = BA = LI
Φ_e = (1/(4πε₀))(2/3)q²a²/c³
Φ_e = IV = IR² (E dissipated by I)
Φ_E = q/ε₀
γ = 3e²λ/(π²k²T) = J/E = ρ⁻¹
G = 1/R
H = B/μ
I = q/t = V/R
I = qv/(2πr) (I loop)
J = B − μ₀H
Λ = R⁻¹
m_EM = (2/3)μ(q²/r)
mv = qrB (q in B)
μ = B/H
μ = IA = qL/(2m) = qvr/2 (I loop)
μ = μ₀μ_r
μ_r = μ/µ₀
M = (B/μ₀) − H
M_e = (1/(4πε₀))(q²/(4πc³))(a/r)²sin²(θ)
p = Bqr
p = qr
P = 2πm/(Bq)
P = D – (ε₀E)
P/2 = πm/(Bq) = πr/v (q in B)
q = CV (capacitor)
q = ΔΦ/R
q = It
r = mv/(Bq) (q in B)
ρ = Rσ/l (wire)
ρ_E = ½εE²
ρ_E = ½μH²
R = F_m/Φ
R = V/I
V = (1/(4πε₀))(q/r) = E_PE/q = IR = K(q/r)
Y = G + Bi
Y² = G² + B²
Ψ = DA
Z = R + Xi
Z = E/H = √(μ/ε) = V/I
abs(Z) = √(R² + X²)

(1/n_f²) − (1/n_i²) = (n_i² − n_f²)/(n_i²n_f²)
C = nλ = nh/p
ΔE = gμ_BB (between +½ & −½ in B)
E = −μB (E of interaction with B)
E_i = eV_i
E_iK = R_H(Z − 1)²
E_K = ½mv² = Ze²/((4πε₀)2r)
E_n = −½(K_eq₁q₂/r) = ½mv² + −(K_eq₁q₂/r) = ½mv_n² = ½(p_n²/m) = −2π²K²mq⁴/(n²h²) = −(m/(2h²))(Ze²/(4πε₀))²(1/n²) = −me⁴/(8h²ε₀²n²) = −R_H/n² = −R_H(Z²/n²)
E_PE = −e²/(4πε₀r) = K_eq₁q₂/r = m(n²h²/(4π²r²m²)) = −(Ze²/((4πε₀)r))
E_R = abs(ΔE) = E_i − E_f = E_Xi − E_Xf = (h²/(8mL²))(n_i² − n_f²) = hf = R_H((1/n_f²) − (1/n_i²))
E_X = −(2π²mZ²e⁴/((4πε₀)²h²))(1/n²) = −(Ze²/((4πε₀)2r)) = (Ze²/((4πε₀)2r)) + [−(Ze²/((4πε₀)r))] = −(Zeq/((4πε₀)2r))
f = (1/(2π))[e²/((4πε₀)mr³)]^1/2 = (1/(2π))(v/r) = me⁴/(4ε₀²h³n³) = ω/(2π)
f = (2π²mZ²e⁴/((4πε₀)²h³))((1/n_f²) − (1/n_i²)) = cR((1/n_f²) − (1/n_i²)) = (E_i − E_f)/h = (me⁴/(8ε₀²h³))((1/n_f²) − (1/n_i²)) = (R_H/h)((1/n_f²) − (1/n_i²))
f = 4π²mZ²e⁴/(n³h³(4πε₀)²)
F_c = mv²/r = Ze²/((4πε₀)r^²)
f_Kα = cR_∞(Z − 1)²((1/1²) − (1/2²))
f_L = μ_BB/h = qB/(4πm)
γ = e/(2m)
K = 2π/λ
L = ½(h/(2π)) (intrinsic)
L = m_lh = mvr = nh/(2π) = hr/λ
L² = m²v²r² = n²h² = (1/(4πε₀))(Ze²/r²)mr
λ = 2μZeq²/(h²α)
λ⁻¹ = R_∞((1/n_f²) − (1/n_i²))
m = MN
μ = −γμ_BL/h = IA = m_lqh/(4πm) = lμ_B = −μ_BL/h = −qL/(2m) = qvr/2
p = m_lh/(2π)
p_n = h/λ_n
r = (4πε₀)h²n²/(Ze²m) = a₀n² = n²h²(4πε₀)/(4π²mZe²) = n²h²/(4π²K_emq²) = nh/(mv2π)
v = (1/(4πε₀))(Ze²/(nh)) = fλ = nh/(2πrm) = nh/(mr)

A = A₀e^−λt = λN
A = Z + N
A_r = m/m_u
ΔE = Δmc²
Δm = m_p − m_d − m_α
−ΔN = λNΔt
ΔN/Δt = λN
E_BN = Δmc² = [Zm_p + Nm_n − m_A]c²
E_BN/n_N = [Zm_H⁰ + (A − Z)m_n − m_A⁰]c²/A
E_Kα = Q/(1 + (m_α/m_d)) ≈ ((A_p − 4)/A_p)Q
E_Kd = (½m)(hf/c)²
E_R = E_i − E_f = hf
f = (A_r − A)/A
F_n = ρ_nv_n
λ = −ΔN/(NΔt) = ln(2)/t_1/2
m = m₀(½)^(t/t_1/2) = m₀e^−λt
m_α/m_d ≈ 4/(A_p − 4)
m_αv_α = m_dv_d
μ = g√(s(s + 1))μ_N
N = N₀e^−λt
N/N₀ = e^−λt
q = Ze
Q = ½m_αv_α²((m_α/m_d) + 1) = (½m_d(m_α²/m_d²)v_α²) + (½m_αv_α²) = ½m_dv_d² + ½m_αv_α² = Δmc² = E_Kd + E_Kα = [(m_A + m_B) − (m_C + m_D)]c² = (m_p − m_d − m_e)c² = (m_p − m_d − m_α)c²
r = ³√(A)c (c = 1.2 fm)
ρ_q(r) = ρ_q/(1 + e^{(r − R)/a}) (R = (1.07 × 10⁻¹⁵ ± 2 × 10⁻¹⁷)A^1/3 m & a = 5.5 × 10¹⁴ ± 7 × 10¹³ m)
σ_D-D = (288/E_K[keV])e^{−45.8/√(W[keV])} × 10⁻²⁸ m²
t = −ln(N/N₀)/λ
t_1/2 = ln(2)/λ = τln(2)
τ = λ⁻¹ = t_1/2/ln(2)
v_d = (m_α/m_d)v_α
X = q[esu]²/(10V[cm³]γ[erg/cm²]) ≈ Z²/(47A)

a = (−g/2) − 1 = (abs(μ)/(qsh/m)) − 1

a = γ³a₀

a = L/(mc)
a² + e² < (Gm/c²)²
α = 1 − (r₀/r) + (e²/r²) = 1 − (r₀/r) + (e²/r²) − ((1/3)Λr²) + (r₀/R) + ((1/3)ΛR²) − (e²/R²)
A = 4π(2Gm/c²)² = 4π(r₊² + a²)

a = ccos(θ_ac) (right triangle)
a/c = cos(θ_ac) (right triangle)
a/sin(θ_bc) = b/sin(θ_ac) = c/sin(θ_ab)
A = ½ab (right triangle)
A = ½absin(θ_ab) = ½acsin(θ_ac) = ½bcsin(θ_bc) = bh/2 = √(s(s − a)(s − b)(s − c)) (Heron's formula) (triangle)
A = ½nr²sin(2π/n) (inscribed polygon of n sides, circle)
A = ½s² (isosceles triangle)
A = (1/4)nl²cot(180°/n) (polygon of n sides)
A = (1/4)πd² = πr² (circle)
A = 2A_base + h(a + b + c) = 3lw + bh (triangular prism)
A = 2(lw + lh + wh) (rectangular prism)
A = 2πrh + 2σ = 2πr(h + r) (right circular cylinder)
A = 2πrh (lateral surface area of a right circular cylinder)
A = 4πr² = πd² (sphere)
A = 6l² (cube)
A = bh (parallelogram)
A = h(a + b)/2 (parallelogram/trapezoid)
A = l² (square)
A = l² + 2ll_side (square pyramid)
A = lh = lw (rectangle)
A = nr²tan(π/n) (circumscribed polygon of n sides, circle)
A = πab (ellipse)
A = πrl (lateral surface area of a right circular cone)
A = π(r₁ + r₂)(r₁ − r₂) (annulus)
A = πr² + ½Cl_side = πr√(r² + h²) + σ = πr(r + l_side) = πr(r + √(r² + h²)) (right circular cone)
A = Pa/2 (any regular polygon; a = apothem, the distance from the center of the polygon to the center of one side)
A = (w(2h + 2l)) + 2hl (prism)
A_arc = (θ_arc/360°)πr²

(ab)^m = a^mb^m
a^maⁿ = a^{m + n}
(a^m)ⁿ = a^mn
am(u) = φ

a₀ = 4πε₀h²/(m_ee²) = α/(4πR_∞)
a₀ = 0.3cH
a_P = √(c⁷/(hG)) (Planck acceleration)
A_P = t_P⁻¹
ε₀ = 1/(μ₀c²) (electric constant)
Φ₀ = h/(2e)
Φ² = Μ²/(2Λ) = v²/2 (Higgs field)
Φ_P = m_Pl_P²/(t_P²I_P)
G₀ = 2e²/h (conductance quantum)
G_β = G_Fcos(θ_C)

G_P = t_P³I_P²/(m_Pl_P²)

K_e = 1/(4πε₀)
K_J = 2e/h (Josephson constant)
K_m = μ₀/(4π)

m_c² ≈ m_dm_t
m_e^1/2 = m_d^1/2 − m_u^1/2
m_e^1/2 − m_μ^1/2 = m_d^1/2 − m_s^1/2

mH2 = 2λν2 = 2Μ2
mμ1/2 − me1/2 = ms1/2 − md1/2
mμ3 ≈ memτ2
mn1/2 − mp1/2 = mΞ−1/2 − mΞ01/2 = mΞc01/2 − mΞc+1/2
mνe1/2 − me1/2 = mu1/2 − md1/2

m_s^² ≈ m_um_b


m_τ = (m_μ³/m_e)^1/2
m_tt = m_W⁺ + m_W⁻ + m_W⁰
m_um_s³m_t² ≈ m_dm_c³m_b²





R = N_Ak
R_∞ = α²m_ec/(2h) = 2pi²m_ee⁴/((4πε₀)²h³c) = m_ee⁴/(8ε₀²h³c)
R_H = R_∞hc
R_K = h/e² = μ₀c/(2α) (von Klitzing constant)
R_P = m_Pl_P²/(t_P³I_P²)

ae = (abs(μe)/μB) − 1
α = 2πKee2/(hc) = e2/(2ε0hc) = e2/(4πε0hc) = (e/qP)2 = μ0ce2/(2h) (fine structure constant)
αμ < (1.1/2.3) × 10−8 (between μ-ic γ & μ)
αS(E) = 12π/((33 − 2nf)ln(E2/Λ2))
αW = sin2(θW) = 1 − (mW/mZ)2
αW/αS ≈ √(τΔ+/τΣ+)
αW(mZc2) = gW2/(4π)
αX0 = gX02/(4π)