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JEE Advanced 2023 · Paper 1, Physics question 1

A slide with a frictionless curved surface, which becomes horizontal at its lower end, is fixed on the terrace of a building of height 3h3h from the ground, as shown in the figure. A spherical ball of mass mm is released on the slide from rest at a height hh from the top of the terrace. The ball leaves the slide with a velocity u⃗0=u0x^\vec u_0 = u_0\hat x and falls on the ground at a distance dd from the building making an angle θ\theta with the horizontal. It bounces off with a velocity v⃗\vec v and reaches a maximum height h1h_1. The acceleration due to gravity is gg and the coefficient of restitution of the ground is 1/31/\sqrt3. Which of the following statement(s) is(are) correct?

A building of height 3h with a curved frictionless slide on its terrace; the slide starts a height h above the terrace and becomes horizontal at the terrace edge, where the ball leaves with velocity u₀ along +x. The ball follows a dashed parabola to the ground, landing a distance d from the building at angle θ to the horizontal, then bounces with velocity v and rises to a maximum height h₁. Axes: x to the right, z up; g downward.
Choose one or more options
Answer and worked solution

Official answer: A, C, D

Leaving the slide. The slide is frictionless, so u0=2ghu_0=\sqrt{2gh} and u⃗0=2gh x^\vec u_0=\sqrt{2gh}\,\hat x. (A) true.

Landing. The ball falls 3h3h:

  • vertical speed 6gh\sqrt{6gh};
  • time of flight 6h/g\sqrt{6h/g}.

So tan⁡θ=6gh2gh=3\tan\theta=\dfrac{\sqrt{6gh}}{\sqrt{2gh}}=\sqrt3, giving θ=60∘\theta=60^\circ. (C) true.

Bounce. The vertical component becomes 136gh=2gh\frac{1}{\sqrt3}\sqrt{6gh}=\sqrt{2gh}, upward; the horizontal component is unchanged. So
v⃗=2gh (x^+z^).\vec v=\sqrt{2gh}\,(\hat x+\hat z).
(B) false: the z^\hat z component is upward, not downward.

Ratio.

  • h1=(2gh)22g=hh_1=\dfrac{(\sqrt{2gh})^2}{2g}=h;
  • d=2gh⋅6h/g=23 hd=\sqrt{2gh}\cdot\sqrt{6h/g}=2\sqrt3\,h.

So d/h1=23d/h_1=2\sqrt3. (D) true.

Topics:Projectile motionConservation of mechanical energyElastic and inelastic collisions
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