# Fall Testing of the Personal Airborne Equipment Backpack: Ground and Flight Testing
## 4 Discussion  
The force impact value after the discontinuance of the backpack's free-fall depends on the kinetic energy mass which the backpack is losing and the trajectory from which the loss originates. The force impact value after the discontinuance of the free-fall can be defined as [15]:  
\[F=\frac{E_{k}}{s}=\frac{m\cdot v^{2}}{2\cdot s} \tag{1}\]  
where:  
\(F\)--force impact [N]  
\(E_{k}\)--kinetic energy [J]  
\(s\)--trajectory, on which the retardation of the backpack originates and achieves zero speed [m]  
\(v\)--speed at a time of the retardation starts [m\(\cdot\)s\({}^{-1}\)]  
\(m\)--backpack weight [kg] \(m=28\) kg  
The free-fall speed of the backpack at the moment when the stretched rope turns and starts to retard the backpack can be calculated by a function (valid for the cases with no air resistance, which could be neglected with regard to a 4 m free-fall height):  
\[v=\int_{t_{0}}^{t_{1}}g\cdot dt=g\cdot\int_{t_{0}}^{t_{1}}dt \tag{2}\]  
\[v=g\cdot[t]_{t_{0}}^{t_{1}}=g\cdot[t_{1}-t_{0}] \tag{3}\]  
where:  
\(v\)--speed at a time of the retardation starts [m\(\cdot\)s\({}^{-1}\)]  
\(g\)--gravitational acceleration [m\(\cdot\)s\({}^{-2}\)] \(g\) = 9.81 m\(\cdot\)s\({}^{-2}\)  
\(t\)--free-fall time [s]  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Time [s]** & **Force [N]** & **Time [s]** & **Force [N]** & **Time [s]** & **Force [N]** & **Time [s]** & **Force [N]** & **Time [s]** & **Force [N]** \\ \hline
0 & 20 & 0.9 & 1501 & 1.8 & 500 & 2.7 & 294 & 3.6 & 245 \\
0.05 & 10 & 0.95 & 608 & 1.85 & 255 & 2.75 & 45  
If \(t_{0}=0\) then \(t=t_{1}\) and the free-fall speed of the backpack at the moment when the rope gets stretched and begins to retard the backpack, then we can calculate by a familiar function (valid for cases with no air resistance, which could be neglected with regard to a 4 m free-fall height):  
\[v=g\cdot t \tag{4}\]  
The free-fall period of the backpack anchored by a 4 m rope was experimentally discovered during the fall tests and its value is 0.65 s. The trajectory on which the retardation of the backpack starts and achieves zero-value speed was determined by following the rope producer's declaration of a maximum extension of up to 5%. Since the free-fall of the backpack can be discontinued by the 4 m long rope, then the retardation starts from the 0.2 m section. (This is relevant for fall tests numbers 1, 2, and 3, which were performed from a fixed point). In the case of free-fall test number 4, a part of the kinetic energy was absorbed by the polyamide strap of the carrying gear of the parachute set OVP-80. Thus, based on an expert predetermination, we can estimate that the retardation started at the 0.3 m section. Based upon the presented relations and assumptions, the value of the calculated force impact during the fall tests of the backpack from the fixed point \(F_{1}\) (retardation on the 0.2 m section) is:  
\[F_{1}=\frac{E_{k}}{s}=\frac{m\cdot v^{2}}{2\cdot s}=\frac{m\cdot\left(g\cdot t \right)^{2}}{2\cdot s}=\frac{28\times\left(9.81\times 0.65\right)^{2}}{2 \times 0.2}=2846\text{ N} \tag{5}\]  
The value of the calculated force impact during the fall tests of the backpack from the flexible point (retardation on the 0.3 m section):  
\[F_{2}=\frac{E_{k}}{s}=\frac{m\cdot v^{2}}{2\cdot s}=\frac{m\cdot\left(g\cdot t \right)^{2}}{2\cdot s}=\frac{28\times\left(9.81\times 0.65\right)^{2}}{2 \times 0.3}=1897\text{ N} \tag{6}\]  
Maximal force impact values, measured at the time of the fall in test numbers 1, 2, and 3 (average value 3590 N) are similar to the calculated force impact value of \(F_{1}=2846\) N. Moreover, the maximum force impact value measured during fall test number 4 (2080 N) is similar to the calculated force impact value \(F_{2}=1879\) N. The values measured in this way can be considered correct with the accepted simplification that the length of the rope and its calculated elongation were used in the calculations. It would be more accurate to use a center-of-gravity shift in the calculations.