A girl throws a ball with initial velocity v at an  inclination of 45°. The ball strikes the smooth vertical wall at a horizontal distance d from girl and after rebounding returns to her hand. What is the coefficient of restitution between wall and ball ?

See fig.As the wall is smooth, the vertical component of the impulse it receives is zero, The total time of flight is given by0=vsin⁡45∘t−12gt2 or t=2vsin⁡45∘gThe time of flight t2 from G to A=dvcos⁡45∘The time of flight t2 from A to G=devcos⁡45∘where e = coefficient of restitution. So, 2v sin⁡45∘g=dVcos⁡45∘+devcos⁡45∘or 2vg=d2v+d2ev vg=dV(1+1e) or v2gd=1+1e Solving, we get e=gdv2−gd.