How Far a Load Can Survive: A Combined Bound from Jolting, Time, and Temperature Rise

Mechanical damage alone gives a misleading answer to the question of how far fragile produce can be carried, because the gentlest mode is usually the slowest, and slowness exposes the load to heat for longer. This paper combines three effects into a single survival bound for firm-ripe tomatoes picked at 24 °C: jolting, which accumulates per kilometre; elapsed time; and a rising load temperature, which accelerates quality loss. The bound requires the surviving saleable fraction S(d) = exp(-k d - ∫₀d/v r(T(t))\,dt) to remain at or above 0.80. Solved numerically, the maximum viable distances on paved road are 10.67 km walking, 11.90 km cycling, and 4.40 km by cart. On gravel they are 8.33 km walking, 7.29 km cycling, and 2.86 km by cart. On rough earth they are 5.96 km walking, 4.33 km cycling, and 1.77 km by cart. The combined bound also produces a reversal: cycling leads on paved road, where its speed shortens heat exposure, while walking leads on gravel and rough earth, where the bicycle's higher jolting rate outweighs its speed. For the walking carrier on paved road, heat and time account for 0.618 of the total loss exponent at the maximum distance, so a jolt-only calculation overstates the walking range by more than half.