JEE Advanced 2023 — Paper 1 — PHY-I solution

JEE Advanced2023Paper 1PHY-I

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 $u_{0} = u_{0} \overset{x}{^}$ 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 $v$ and reaches a maximum height $h_{1}$ . The acceleration due to gravity is $g$ and the coefficient of restitution of the ground is $1/ 3$ . Which of the following statement(s) is(are) correct?

A
$u_{0} = 2 g h \overset{x}{^}$
B
$v = 2 g h (\overset{x}{^} - \overset{z}{^})$
C
$θ = 6 0^{\circ}$
D
$d / h_{1} = 23$

Solution

Velocity at the bottom of the slide. Energy conservation on the frictionless slide from rest at height $h$ gives $u_{0} = 2 g h$ , directed along $+ \overset{x}{^}$ . Hence $u_{0} = 2 g h \overset{x}{^}$ , so option (A) holds.

Velocity just before impact. Horizontal component is unchanged: $v_{x} = 2 g h$ . Vertical drop is $3 h$ , so $v_{z} = - 2 g (3 h) = - 6 g h$ .

Angle of incidence. $tan θ = \frac{∣ v _{z} ∣}{v _{x}} = \frac{6 g h}{2 g h} = 3 \Rightarrow θ = 6 0^{\circ}$ . Option (C) holds.

Bounce. Horizontal component is preserved (smooth ground); vertical component reverses and scales by $e = 1/ 3$ : $v_{z}^{'} = \frac{1}{3} 6 g h = 2 g h$ . So $v = 2 g h \overset{x}{^} + 2 g h \overset{z}{^}$ , which does not match option (B).

Maximum height after bounce. $h_{1} = \frac{( v _{z}^{'} ) ^{2}}{2 g} = \frac{2 g h}{2 g} = h$ .

Horizontal range from building base. Time of flight $t = \frac{2 ( 3 h )}{g} = \frac{6 h}{g}$ , so $d = v_{x} t = 2 g h \cdot \frac{6 h}{g} = 23 h$ .

$∴ \frac{d}{h _{1}} = \frac{2 3 h}{h} = 23$ . Option (D) holds.

Correct options: (A), (C), (D).

← Back to full paper

Practice more PHY-I

Concept-wise practice with instant solutions on Fundamenthol.

Start practicing →