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 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
Updated On: Aug 16, 2023
- \(v=\sqrt{2gh}\,\,\hat{x}\)
- \(\theta=60^{\circ}\)
- \(\vec{v}=\sqrt{2gh}\,(\hat{x}-\hat{z})\)
- \(\frac{d}{h_1}=2\sqrt3\)
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The Correct Option is A, B, D
Solution and Explanation
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.
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.
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Concepts Used:
Rotational Motion
Rotational motion can be defined as the motion of an object around a circular path, in a fixed orbit.
Rotational Motion Examples:
The wheel or rotor of a motor, which appears in rotation motion problems, is a common example of the rotational motion of a rigid body.
Other examples:
- Moving by Bus
- Sailing of Boat
- Dog walking
- A person shaking the plant.
- A stone falls straight at the surface of the earth.
- Movement of a coin over a carrom board
Types of Motion involving Rotation:
- Rotation about a fixed axis (Pure rotation)
- Rotation about an axis of rotation (Combined translational and rotational motion)
- Rotation about an axis in the rotation (rotating axis)