Impulse-Momentum Equation & Its Applications Notes

Last reviewed 16 Sept 2026 · 6 min read

Impulse-momentum principle

Newton's second law applied to a flowing fluid: the net external force on the fluid in a control volume equals the rate of change of momentum of the fluid passing through it.

FormulaMomentum equation (steady flow, one inlet and one outlet)

\sum F_{x} = ρQ (V_{2 x} - V_{1 x}), \sum F_{y} = ρQ (V_{2 y} - V_{1 y})

Impulse form: $F d t = d (mV)$ .

$\sum F$ includes pressure forces on the inlet/outlet sections, gravity (weight of fluid) and the force from the solid boundary. The force exerted by the fluid on the boundary is equal and opposite.

The momentum equation is a vector equation — resolve into components.
Unlike Bernoulli's equation, it needs no information about internal losses; it is therefore used where losses are unknown (hydraulic jump, sudden enlargement, jets).
For non-uniform velocity, multiply momentum flux by the momentum correction factor $β$ (4/3 laminar pipe flow; ≈1.0–1.04 turbulent).

Force on pipe bends and reducers

For a bend turning the flow through angle $θ$ (horizontal plane), with inlet pressure $p_{1}$ , area $A_{1}$ , velocity $V_{1}$ and outlet $p_{2}$ , $A_{2}$ , $V_{2}$ , the force exerted by the fluid on the bend:

FormulaPipe bend

F_{x} = ρQ (V_{1} - V_{2} cos θ) + p_{1} A_{1} - p_{2} A_{2} cos θ

F_{y} = - (ρQ V_{2} sin θ + p_{2} A_{2} sin θ)

F = F_{x}^{2} + F_{y}^{2}

For a reducer (straight, $θ$ = 0): $F_{x} = ρQ (V_{1} - V_{2}) + p_{1} A_{1} - p_{2} A_{2}$ .
These forces are resisted by anchor (thrust) blocks at bends, tees, dead ends and reducers in water mains.

Force exerted by a jet

Jet of area $a$ , velocity $V$ , density $ρ$ .

Stationary plates

Plate	Force in the direction of the jet
Flat plate normal to the jet	$F = ρ a V^{2}$
Flat plate inclined at $θ$ to the jet	Normal to plate: $F_{n} = ρ a V^{2} sin θ$ ; along jet: $F_{n} sin θ$
Curved plate, jet at the centre (symmetric), deflected through angle $(180° - θ)$	$F = ρ a V^{2} (1 + cos θ)$
Hemispherical cup (jet turned back through 180°)	$F = 2 ρ a V^{2}$
Curved plate, jet striking at one tip (unsymmetrical, tangential entry)	$F_{x} = ρ a V^{2} (1 + cos θ)$

Moving plates

If the plate moves in the jet direction with velocity $u$ , the relative velocity is $(V - u)$ :

Case	Force	Work done per second
Single flat plate moving away	$ρ a (V - u)^{2}$	$ρ a (V - u)^{2} u$
Series of flat radial plates (wheel) — the whole jet is used	$ρ aV (V - u)$	$ρ aV (V - u) u$
Series of hemispherical (curved) vanes	$2 ρ aV (V - u)$	$2 ρ aV (V - u) u$

FormulaEfficiency of a series of vanes

Flat radial vanes: $η = \frac{2 u ( V - u )}{V ^{2}}$ → maximum 50% at $u = V /2$ .
Hemispherical vanes (Pelton-type): $η = \frac{4 u ( V - u )}{V ^{2}}$ → maximum 100% (theoretical) at $u = V /2$ .
Single moving flat plate: maximum work at $u = V /3$ ; maximum efficiency $8/27$ ≈ 29.6% (based on kinetic energy of the jet).

Jets on curved vanes with inlet and outlet angles

For turbines, velocity triangles at inlet and outlet give the work done per second per unit weight:

\frac{1}{g} (V_{w 1} u_{1} \pm V_{w 2} u_{2})

( $V_{w}$ = whirl component; + when the outlet whirl is opposite to vane motion). Detailed in Hydraulic Turbines.

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Impulse-Momentum Equation & Its Applications

Impulse-momentum principle

Force on pipe bends and reducers

Force exerted by a jet

Stationary plates

Moving plates

Jets on curved vanes with inlet and outlet angles

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