This lab unifies the delivered scenarios: driven cart, braking, rolling down a chute, sled on a slope and level cart with a constant resistive force.

m(1+κ)·dv/dt = m g sinθ + F − [μ(v) m g cosθ + R] − c v − k v|v| [N]; x measured down the slope.

Delivered special cases: m dv/dt = F − c v → v(t) = F/c + (v₀ − F/c)·e^(−ct/m); braking: v(t) = max(0, v₀ − μ g t), s_stop = v₀²/(2μg); rolling: a = g sinθ/(1+κ), x = ½at²; sled: a = g(sinθ − μ cosθ); cart: a = (F − R)/m.

Extensions added here: quadratic drag k v|v|; a static-excess friction μ(v) = μ[1 + (μs/μk − 1)e^(−|v|/0.5 m/s)] that holds a body at rest unless the push exceeds μs·N; a brake/reaction delay before friction acts (cycling preset); and the energy ledger W_g + W_F = ΔKE + Q_friction + Q_drag.

Honest simplifications. A point body on a straight, rigid slope. Rolling is represented by the inertia factor κ = I/(m r²) with μ acting as a rolling-resistance coefficient — slip, wheel contact patch and tyre deformation are not modelled. The Stribeck excess uses a fixed 0.5 m/s velocity scale. Drag is a lumped coefficient, not a shape-based estimate. Preset values and captions are those of the original simulations.

The closed-form dashed line is drawn whenever quadratic drag and the static excess are off.

Slope, Sliding, Rolling and Braking: Force Balance with Drag Case-Study Lab

Graph