## 3 Some Simple One-Dimensional Transport Scenarios
### Start-Stop Scenarii  
The \"fly-through\" scenario discussed above is an artificial one, only invented for motivating the entity of physical action as being a useful figure of merit for the hitherto undefined property of transport value. Realistic transport scenarios distinguish from this highly idealized case in that usually, a mass \(M\) starts out at point \(A\) from rest, speeds up to a finite speed \(v_{ab}>v_{AB}=L_{AB}/\tau_{AB}\) and slows down upon approach to its endpoint \(B\), where it comes to rest again. In the following sections, we will call such point-to-point transport scenarios \"start-stop transports\". Figure 4 displays the timing of the acting acceleration and deceleration forces across the entire transport duration \(\tau_{AB}\). For \(\tau_{AB}\) to match the duration of the \"fly-through transport\" discussed in Section 3.1, accelerations \(a\) and intermediate velocities \(v_{ab}\) need to be adapted to the chosen lengths of the acceleration and deceleration phases \(\delta\):  
\[a(L_{AB},\tau_{AB},\delta)=\left[\frac{L_{AB}}{\tau_{AB}\delta}\right]\frac{1} {1-\frac{\delta}{\tau_{AB}}} \tag{6}\]  
and  
\[v_{ab}(L_{AB},\tau_{AB},\delta)=\left[\frac{L_{AB}}{\tau_{AB}}\right]\frac{1} {1-\frac{\delta}{\tau_{AB}}}. \tag{7}\]  
Figure 4: Acceleration and deceleration phases in start–stop scenarios reaching out to a transport length \(L_{AB}\) in time \(\tau_{AB}\). In Figure 5a, we show acceleration (red) and speed profiles (green), as well as the fractional distance \(s(t)/L_{AB}\) covered up to time \(t\) (blue), all as functions of the fractional transport time \(t/\tau_{AB}\). In this first example, we assumed \(\delta=0.2\tau_{AB}\). Figure 5b, on the other hand, shows the engine and brake power profiles (red) delivered to the mass \(M\) close to points \(A\) and \(B\), as well as the resulting kinetic energy profiles \(T(t)\) (green), both plotted once again against the fractional transport time \(t/\tau_{AB}\). Also shown for comparison is the physical action \(W(t,\delta)\) (blue) attained after time \(\tau_{AB}\) and normalized to the physical action \(W_{AB}(M,L_{AB},\tau_{AB})\) that had been attained at the end of the \"fly-through\" scenario in Section 3.1 above. Figure 6a,b, in turn, plot the same quantities as above, however, for different lengths of acceleration and deceleration times. Figure 6a shows that longer acceleration and deceleration times lead to increasingly curvier distance-versus-time functions, with all functions increasing towards the same finally travelled distances. Figure 6b, on the other hand, shows that increasingly elongated acceleration and deceleration periods lead to increasingly larger final values of transport value \(W_{AB}\). The considerations in Equations (8)-(10) show that this effect is related to the increasingly higher energy demands when \(\delta^{\prime}\)s are increased. Figure 5: (**a**) (Red) acceleration/deceleration profile giving rise to a start-stop transport scenario; (green) velocity and (blue) distance profiles resulting from the impressed acceleration profile; (**b**) (red) acceleration and braking powers delivered to the transported mass \(M\); (green) kinetic energy and (blue) transport value of mass \(M\), again normalized to the transport value of an ideal “fly-through scenario”. Figure 6: (**a**) Fractional progress in transported length \(x(\delta)/L_{AB}\) as a function of fractional transport time \(t/\tau_{AB}\) as evaluated for different durations \(\delta\) of acceleration and deceleration phases; (**b**) \(\delta\)-dependent gains in physical action \(W_{AB}(L_{AB},\tau,\delta)\) relative to the gain \(W_{AB}\) attained in a “fly-through scenario”.