Check accessible motion before reporting a speed

Choose the conserved account

For a fixed electrostatic source and a point-model body with no other work, K + qV is conserved. Therefore K(x) = K0 − q[V(x) − V(0)]. In the simulator K is in eV and charge is in elementary-charge units, so the product of the numerical charge and potential in volts gives an energy in eV. A negative candidate K does not represent negative kinetic energy; it excludes that location from the chosen motion.

Separate a turning point from an endpoint

The supplied potential V = 3(2x − x²) V on numerical x in 0..2 m has a maximum of 3 V at x = 1 m. For positive charge, K0 below the barrier gives a turning point before the maximum. Solving K = 0 selects xturn = 1 − sqrt[1 − K0/(3q/e)] for the initial forward branch. The other algebraic root lies across a forbidden region and cannot be reached by this classical initial condition.

Optional threshold-time enrichment

At exact positive-charge barrier energy, K is proportional to (x − 1)² near the top. Since speed is proportional to sqrt(K), travel time contains integral dx/|x − 1| and diverges. Thus nonnegative K everywhere is not alone sufficient to certify finite-time crossing of this exact threshold. This logarithmic diagnosis supplements the core energy balance; no quantum tunneling or real species behavior is modeled.

Reverse charge, not the potential

A negative charge moves toward lower potential energy when V increases. In this packet its candidate K grows toward the potential maximum, so the same positive potential hill is not an energy barrier for that charge. The hypothetical mass only converts K into speed after accessibility is established.

Worked original case

For q = +e and K0 = 2 eV, xturn = 1 − sqrt(1/3) ≈ 0.422650 m. For q = +e and K0 = 3 eV, the top is an asymptotic limit, not a finite-time crossing. For q = −e and K0 = 2 eV, K at x = 1 m is 5 eV.

Investigation

Compare positive, zero and negative charge at fixed initial K. Predict the minimum candidate energy and accessible branch before plotting. Retain one ordinary turning point, one threshold case and one crossing case; state why their endpoint criteria differ.

Changed-case practice

Sources and evidence boundary

Original teaching and cases checked 30 October 2026 against the current official framework, Unit 9 and OpenStax electric potential. All records are supplied or generated, not measured by a learner. No physical charging, battery connection, high-voltage equipment or particle experiment is requested. Retain predictions, derivations and critique in the course evidence workspace; public feedback is not independent mastery or performed laboratory time.