A slide with a frictionless curved surface, which becomes horizontal at its lower end, is fixed on the terrace of a building of height 3h from the ground, as shown in the figure. A spherical ball of mass mis released on the slide from rest at a height hfrom the top of the terrace. The ball leaves the slide with a velocity u→0=u0x∧and falls on the ground at a distance dfrom the building making an angle θwith the horizontal. It bounces off with a velocity v→and reaches a maximum height h1. The acceleration due to gravity is gand the coefficient of restitution of the ground is 13. Which of the following statement(s) is(are) correct?

JEE Advanced · Physics · 4. Motion in 2D

A slide with a frictionless curved surface, which becomes horizontal at its lower end, is fixed on the terrace of a building of height $3 h$ from the ground, as shown in the figure. A spherical ball of mass $m$ is released on the slide from rest at a height $h$ from the top of the terrace. The ball leaves the slide with a velocity ${\vec{u}}_{0} = u_{0} \overset{\land}{x}$ and falls on the ground at a distance $d$ from the building making an angle $θ$ with the horizontal. It bounces off with a velocity $\vec{v}$ and reaches a maximum height $h_{1}$ . The acceleration due to gravity is $g$ and the coefficient of restitution of the ground is $\frac{1}{\sqrt{3}}$ . Which of the following statement(s) is(are) correct?

A ${\vec{u}}_{0} = \sqrt{2 g h} \overset{\land}{x}$
B $\vec{v} = \sqrt{2 g h} (\overset{\land}{x} - \overset{\land}{z})$
C $θ = 60^{o}$
D $\frac{d}{h_{1}} = 2 \sqrt{3}$

Verified Solution

Answer & Solution

Correct Answer

(A) ${\vec{u}}_{0} = \sqrt{2 g h} \overset{\land}{x}$

Step-by-step Solution

Detailed explanation

Using energy conservation,

m g h = \frac{1}{2} m {(u_{0})}^{2}

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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