Tag

Potential energy

The potential energy term will only include the conservative force of gravitation. To compute it, we need to be able to formulate the height of the trike above sea level in terms of ss. If we have inclination/declination angle data for the hill in question, let us define θ(s)\theta(s) as the declination angle (positive when downhill) at ss metres from the summit. If we form an infinitesimal triangle on the hill/mountain, we will see that dh=−dssin⁡θdh = -d s \sin{\theta} and hence:

dhds=−sin⁡θ(s).\begin{aligned} \dfrac{dh}{ds} &= -\sin{\theta(s)}. \end{aligned}

Integrating from the summit, we obtain that:

h(s)=hsummit−∫0ssin⁡θ(σ)dσ.\begin{aligned} h(s) &= h_{\mathrm{summit}} - \int_{0}^s \sin{\theta(\sigma)}d\sigma. \end{aligned}

Where hsummith_{\mathrm{summit}} is the height of the hill/mountain, or rather its tallest point accessible by road, and we use σ\sigma as a dummy (integration) variable for distance from the summit.

Hence if gravitational acceleration can be assumed to be constant:

V=mg(hsummit−∫0ssin⁡θ(σ)dσ).\begin{aligned} V &= mg\left(h_{\mathrm{summit}} - \int_0^s \sin{\theta(\sigma)}d\sigma\right). \end{aligned}
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Last modified: October 23, 2026.