Equation desk

Use an equation only after naming the system, the variables, and the approximation that made it true. The unit checks below are compact tests, not substitutes for derivation.

Motion and force

v = dx/dt defines velocity as the local slope of position x(t); units are m/s. Differentiating again gives a = dv/dt, in m/s². Newton’s law is ΣF = ma, with N = kg·m/s².

For constant acceleration, integrate dv = a dt to get v = v₀ + at, then integrate dx = v dt to get x = x₀ + v₀t + ½at². The ½ is the average of the linearly changing velocity contribution.

Work, energy, and power

The differential transfer is dW = F · dr; integration gives work in joules. The work theorem follows from F = ma and a dx = v dv, giving W_net = Δ(½mv²). Gravitational energy follows from dU = −F·dr, so near Earth ΔU = mgΔh.

Power is energy rate: P = dW/dt = F · v. For a water stream, ṁ = ρQ and lifting power is P_h = ρgQH. Since ρQ is kg/s, gH is J/kg, and the product is J/s.

Pressure and flow

Pressure is normal force per area: p = F_n/A, in pascals. Hydrostatic balance on a thin slab gives dp/dz = −ρg, so constant-density depth h produces gauge pressure Δp = ρgh.

Steady incompressible continuity is Q = Av. The extended Bernoulli balance is

p/ρg + v²/2g + z + h_p = constant + h_L.

Every term is metres of head. h_p adds mechanical energy per unit weight; h_L removes it.

Viscous loss and scaling

For fully developed laminar flow in a circular pipe, integrating 0 = −dp/dx + μ(1/r)d/dr(r du/dr) gives Q = πR⁴Δp/(8μL). The fourth power of radius explains why a small blockage can dominate resistance. Use it only when the flow is Newtonian, laminar, steady, and fully developed.

The Reynolds number Re = ρvD/μ compares inertia with viscosity. It is dimensionless because both numerator and denominator reduce to kg/(m·s). A regime estimate is useful only with geometry and boundary conditions stated.

Symbols and SI units

• ρ: density, kg/m³; μ: dynamic viscosity, Pa·s; ν = μ/ρ: m²/s.
• Q: volume flow, m³/s; ṁ = ρQ: mass flow, kg/s; A: area, m².
• p: pressure, Pa; H or h: head or height, m; g: 9.81 m/s².
• F: force, N; E or U: energy, J; P: power, W.

Test before trusting

Set a variable to zero, take a large or small limit, and reduce every term to units. If Q → 0, velocity and hydraulic power should vanish; if pipe radius doubles in laminar flow at fixed pressure drop, the predicted flow should increase by sixteen. A formula that fails those tests is being used outside its model.