A ball of mass m moving with speed v strikes a rigid surface. Match...

When a ball strikes a rigid surface at an angle $\theta$ with the normal and rebounds perfectly (elastically) with the same speed $v$ at the same angle, the change in momentum is only along the normal direction.

The magnitude of impulse imparted is given by:
$J = |\Delta p| = 2mv \cos\theta$

(A) Strikes normally ($\theta = 0^\circ$):
$J = 2mv \cos(0^\circ) = 2mv$ (Matches II)

(B) Strikes at $30^\circ$ to the normal ($\theta = 30^\circ$):
$J = 2mv \cos(30^\circ) = 2mv \left(\dfrac{\sqrt{3}}{2}\right) = \sqrt{3}mv$ (Matches III)

(C) Strikes at $60^\circ$ to the normal ($\theta = 60^\circ$):
$J = 2mv \cos(60^\circ) = 2mv \left(\dfrac{1}{2}\right) = mv$ (Matches I)

(D) Strikes at $45^\circ$ to the normal ($\theta = 45^\circ$):
$J = 2mv \cos(45^\circ) = 2mv \left(\dfrac{1}{\sqrt{2}}\right) = \sqrt{2}mv$ (Matches IV)

Therefore, the correct matching is (A)-(II), (B)-(III), (C)-(I), (D)-(IV).

Answer: (A)-(II), (B)-(III), (C)-(I), (D)-(IV)