A ball moving with a velocity v hits a massive wall moving towards the ball with a velocity u. An elastic impact lasts for a time$\Delta \text{t}$

${\text{m}}_{\text{1}}=\text{m,}{\text{m}}_{\text{2}}=\text{M}$

${\text{u}}_{\text{1}}={\text{v}}_{\text{1}}\text{,}{\text{u}}_{\text{2}}=-{\text{u}}_{\text{1}}$

${\text{v}}_{\text{1}}=\frac{{\text{m}}_{\text{1}}-{\text{m}}_{\text{2}}}{{\text{m}}_{\text{1}}+{\text{m}}_{\text{2}}}{\text{u}}_{\text{1}}+\frac{{\text{2m}}_{\text{2}}{\text{u}}_{\text{2}}}{{\text{m}}_{\text{1}}+{\text{m}}_{\text{2}}}$

$=\frac{\text{m}-\text{M}}{\text{m}+\text{M}}\text{v}+\frac{\text{2Mu}}{\text{m}+\text{M}}$

$=-\text{v}-\text{2u}$

$\text{F}=\frac{\text{m}\left({\text{v}}_{\text{1}}-{\text{u}}_1\right)}{\text{t}}=\frac{\text{m}\left(-\text{v}-2\text{u}-\text{u}\right)}{\text{t}}$

$=\frac{\text{m}\left(-2\text{v}-2\text{u}\right)}{\text{t}}$

$=\frac{-2\text{m}\left(\text{u}+\text{v}\right)}{\text{t}}$

$\text{K}\text{.E}=\frac{1}{2}{\text{mv}}_{\text{1}}^{\text{2}}-\frac{1}{2}{\text{mu}}_{\text{1}}^{\text{2}}$

$=\frac{1}{2}\text{m}{\left(\text{v}+2\text{u}\right)}^2-\frac{\text{1}}{\text{2}}\text{m}{\left(\text{v}\right)}^{\text{2}}$

$=2\text{mu}\left(\text{u}+\text{v}\right)$