A point mass m collides with a block of mass 4 m connected with string of length $ℓ$ as shown in the diagram. The collision is elastic in nature and the block deflects to an angle 60° after the collision. The collision is along the line joining the center of the two objects. Mark the CORRECT option(s) :-

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After the collision the block swings to 60°: $\frac{1}{2} (4 m) u^{2} = 4 m g ℓ (1 - cos 6 0^{\circ}) \Rightarrow u = g ℓ$ .

The string constrains the block to move horizontally, so during the collision the tension impulse cancels the vertical (along-string) component of the contact impulse J. With the collision along the 45° line: $u = \frac{J c o s 4 5 ^{\circ}}{4 m}$ , the mass keeps moving along the 45° line, and energy conservation gives rebound $v_{1} = - \frac{7 v _{0}}{9}$ , $J = \frac{16}{9} m v_{0} = 42 m g ℓ$ , $v_{0} = \frac{9}{4} 2 g ℓ$ .

So (A) should be $\frac{9}{4} 2 g ℓ$ - wrong; tension impulse $J sin 4 5^{\circ} = 4 m g ℓ$ - (B) correct; normal impulse is $4 m 2 g ℓ$ - (C) wrong; rebound is $7 v_{0} /9$ - (D) wrong.

Ans. (B)

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