A body of mass m moving with a velocity $\mathrm{v} \mathrm{in} \mathrm{the} \mathrm{x}$ direction collides with another body of mass $\mathrm{M}$ moving in y direction with a velocity $\mathrm{V}$. They coalesce into one body during collision

1) $\mathrm{mv} = (\mathrm{M} + \mathrm{m})\mathrm{V}\text{'} \mathrm{cosq}$

2) $\mathrm{Mv} = (\mathrm{m}+\mathrm{M})\mathrm{V}\text{'}\mathrm{sinq}$

$(\mathrm{m}+\mathrm{M})\mathrm{V}\text{'} =\sqrt{\left(\mathrm{mv}\right)² + \left(\mathrm{MV}\right)²}$

2) Initial kinetic is ; ${\mathrm{k}}_1=\frac{1}{2}{\mathrm{mv}}^2+\frac{1}{2}{\mathrm{MV}}^2$

Final K.E. is ${\mathrm{k}}_{\mathrm{f}}=\frac{1}{2}\left(\mathrm{m}+\mathrm{M}\right)\mathrm{v}{\text{'}}^2$

Decrease in $\mathrm{K}.\mathrm{E} = {\mathrm{k}}_1-{\mathrm{k}}_{\mathrm{f}} ; ∆\mathrm{k}=\frac{\mathrm{mM}}{2\left(\mathrm{m}+\mathrm{M}\right)} (\mathrm{v}² +\mathrm{V}²)$ 

Fraction of initial kinetic transformed into heat is

$\frac{∆\mathrm{k}}{{\mathrm{k}}_1}=\frac{\mathrm{mM}}{\mathrm{m}+\mathrm{M}}\left(\frac{{\mathrm{v}}^2+{\mathrm{V}}^2}{{\mathrm{mv}}^2+{\mathrm{MV}}^2}\right)$