A sphere moving with velocity v strikes elastically with a wall moving towards the sphere with velocity u. If the mass of the wall is infinitely large, the work done by the wall during collision will be

Work done by wall = change in kinetic energy

$=\text{K}{\text{.E}}_{\text{1}}-\text{K}{\text{.E}}_{\text{2}}$

$=\frac{1}{2}{\text{mv}}^{\text{2}}-{\text{mv}}_{\text{1}}^{\text{2}}$

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

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

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

Work done $=\frac{1}{2}{\text{mv}}^{\text{2}}-\frac{1}{2}{\text{mv}}_{\text{1}}^{\text{2}}$

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

$=\frac{1}{2}{\text{mv}}^{\text{2}}-\frac{1}{2}\text{m}\left({\text{v}}^{\text{2}}+4{\text{u}}^{\text{2}}+4\text{vu}\right)$  

${\text{work}}_{\text{1}}=\frac{1}{2}{\text{mv}}^{\text{2}}-\frac{1}{2}\text{m}\left({\text{v}}^{\text{2}}+4{\text{u}}^{\text{2}}+4\text{vu}\right)$

${\text{work}}_{\text{1}}=\frac{1}{2}{\text{mv}}^{\text{2}}-\frac{1}{2}{\text{mv}}^{\text{2}}-\frac{1}{2}\text{m}\left(4{\text{u}}^{\text{2}}\right)-\frac{1}{2}\text{m}\left(\text{4vu}\right)$

${\text{work}}_{\text{1}}=-2{\text{mu}}^{\text{2}}-2\text{mvu}$

${\text{work}}_{\text{1}}=-2\text{mu}\left(\text{u}+\text{v}\right)$

$(-)$ Indicates K.E gained