$ J = \int_{t_{i}}^{t_{f}}F\mathrm{d}t \implies J= \int_{t_{i}}^{t_{f}}ma\mathrm{d}t \implies J = m \int_{t_{i}}^{t_{f}} \frac{\mathrm{d}v}{\mathrm{d}t}\implies J = m(v_{f}-v_{i}) $ $ \begin{align} m_{1}v_{1} + m_{2}v_{2} &= m_{1}v_{1}'+m_{2}v_{2}' \\ \end{align} $ $ m_{1}v_{i} = (m_{1}+m_{2})v_{f} $ $ v_{f} = \frac{m_{1}v_{i}}{m_{1}+v_{2}} $ $ \Delta KE = \left| \frac{1}{2}m_{1}v_{1}^2 - \frac{1}{2}(m_{1}+m2)v_{2}^2 \right| $ $ $