Water-Powered Incline

What is modelled

Two carriages of equal empty mass hang on one rope over a drum, so one goes up as the other comes down and gravity only sees the cargo difference. A water wheel supplies traction at constant power, which means the force falls as speed rises above a reference speed. Rolling resistance, a brake and viscous drag oppose the motion. The page integrates the equation of motion with the 3D page constants (220 m track, 22 m rise, 9 t empty carriages) and keeps a work and energy account.

Equations

• Water power P = ρ g Q H η, traction Fd = P / max(vref, |v|) with v_ref = 0.38 m/s.
• Gravity Fg = (mB - mA) g sinθ with sinθ = 22/220.
• Rolling cr (mA + mB) g cosθ with c_r = 0.009, brake up to 48 kN and drag 14 kN per m/s.
• Equation of motion (mA + mB) dv/dt = Fd + Fg - Fresist.

Checks run in the page

• Sum of the five work terms against the kinetic energy at the end of the trip.
• Gravity work summed step by step against (m_B - m_A) g sin(theta) x distance.
• Long-track settling speed against the bisection root of the force balance.
• Water volume against flow x time, and the energy offered to the wheel.

Honest simplifications

A reduced-order teaching model: constant overall efficiency, a rigid rope with no stretch or wheel inertia, no start-up water delay, a constant slope, and carriages of equal empty mass. The numbers are those of the 3D teaching page, not survey data of the real canal.

• Press Loaded ascent and compare the trip time with Heavy load, weak water, where the carriage may not move at all.
• Press Controlled descent and see the brake absorb the energy that gravity delivers.
• Raise the cargo and read the water used per trip: the extra weight must be paid for by the water wheel.