During head on elastic collision, two bodies of masses ${\text{m}}_1$ and ${\text{m}}_2$ completely exchange their velocities when

Let the body of mass m₁ moving with velocity u₁ moves with velocity v₂ after collision with the body of mass m₂
 which was moving with velocity u₂ and which after collision attains a velocity v₂.

$\therefore {\mathrm{v}}_1=\frac{\left({\mathrm{m}}_1-{\mathrm{m}}_2\right)}{{\mathrm{m}}_1+{\mathrm{m}}_2}{\mathrm{u}}_1+\frac{2{ \mathrm{m}}_2}{{ \mathrm{m}}_1+{\mathrm{m}}_2}{\mathrm{u}}_2$

And , ${\mathrm{v}}_2=\frac{2{ \mathrm{m}}_1}{{ \mathrm{m}}_1+{\mathrm{m}}_2}{\mathrm{u}}_1-\frac{\left({\mathrm{m}}_1+{\mathrm{m}}_2\right)}{{\mathrm{m}}_1+{\mathrm{m}}_2}{\mathrm{u}}_2$.

$\text{ When }{\mathrm{m}}_1={\mathrm{m}}_2=\mathrm{m}$

${\mathrm{v}}_1=\frac{0}{2 \mathrm{m}}{\mathrm{u}}_1+\frac{2 \mathrm{m}}{2 \mathrm{m}}{\mathrm{u}}_2={\mathrm{u}}_2$ and ${\mathrm{v}}_2=\frac{2 \mathrm{m}}{2 \mathrm{m}}{\mathrm{u}}_1-\frac{0}{2 \mathrm{m}}{\mathrm{u}}_2={\mathrm{u}}_1$.

Thus, the bodies exchange speeds after collision.