Bungee jump

Alice falls a distance $-\Delta z = l$ that corresponds to the maximal length $l$ of the rope. Since her velocity is zero at the lowest point, the potential energy is completly converted into the elastic energy of the rope. This energy conservation can be used to calculate the maximum elongation $\Delta l$ of the rope. $\begin{aligned} & & E_\text{pot} &= E_\text{elast} \\ \Rightarrow & & m g l &= \frac{K}{2} \Delta l^2, \quad \Delta l = l - l_0 \\ \Rightarrow & & 0 &= \Delta l^2 - \frac{2 m g}{K} \Delta l - \frac{2 m g}{K} l_0 \\ \Rightarrow & & \Delta l &= \frac{m g}{K} + \sqrt{\left(\frac{m g}{K}\right)^2 + \frac{2 m g}{K} l_0 } \end{aligned}$ We can use the result to calculate the force at the lowest point $\begin{aligned} F &= F_\text{elast} + F_\text{g} \\ &= k \Delta l - m g \\ &= k \sqrt{\left(\frac{m g}{K}\right)^2 + \frac{2 m g}{K} l_0 } \\ \Rightarrow \quad a &= \frac{F}{m} \\ &= \sqrt{1 + \frac{2 K l_0}{m g}} \cdot g\\ &= \sqrt{1 + \frac{2 \cdot 120 \cdot 25}{75 \cdot 10}} \cdot g \\ &= 3 \cdot g \end{aligned}$