Question

A ball is released on a smooth plane as shown. It strikes with the ground, the coefficient of restitution is $\frac{1}{\sqrt{3}}$ Choose the correct options

• A. $v=\sqrt{2gh}\,\,\hat{x}$
• B. $\theta=60^{\circ}$
• C. $\vec{v}=\sqrt{2gh}\,(\hat{x}-\hat{z})$
• D. $\frac{d}{h_1}=2\sqrt3$

Answer 1

Answer: (A), (B), (D)

$v=\sqrt{2gh}\,\,\hat{x}$

Answer 2

The correct options are : (A),(B) and (D).

From energy conservation :
$\frac{1}{2}mv_0^2=mgh\Rightarrow v_0=\sqrt{2gh}$ ………………….Option (A) is correct.

$tan\theta=\frac{v_z}{v_x}=\frac{\sqrt{2g(3h)}}{\sqrt{2gh}}=\sqrt{3}$

$\theta=60^{\circ}$ ……………………………Option (B) is correct.

$\vec{v}=\sqrt{2gh}\,\hat{i}\,-\,\sqrt{2gh(3h)}\,\hat{k}$

$=\sqrt{2gh}(\hat{i}-\sqrt3 \hat{k})$ …………………………….Option(C) is incorrect.

d=$v_0\sqrt{\frac{2(3h)}{g}}=\sqrt{2gh}\sqrt{\frac{2\times3h}{g}}$

d=$2h\sqrt3$

$\frac{d}{h_1}=2\sqrt3$ …………………….. Option (D) is correct.