A small ball of mass m moving with speed v collides elastically with a simple pendulum with bob of mass m at rest. The maximum height attained by the bob after collision is

Options

A. `(v^(2))/(g)`
B. `(v^(2))/(2g)`
C. `(v^(2))/(4g)`
D. 0

Answer

To solve the problem of a small ball of mass \( m \) moving with speed \( v \) colliding elastically with a simple pendulum bob of mass \( m \) at rest, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Collision**:
- The problem states that the collision is elastic. In an elastic collision involving two objects of equal mass, the velocities are exchanged. Therefore, after the collision, the moving ball will come to rest, and the pendulum bob will move with the speed \( v \).

2. **Determine the Speed of the Bob After Collision**:
- After the collision, the speed of the pendulum bob becomes \( v \).

3. **Apply the Conservation of Energy**:
- We will use the principle of conservation of mechanical energy to find the maximum height attained by the pendulum bob. The total mechanical energy before the bob starts moving upward is equal to the total mechanical energy at the maximum height.
- Initially, the pendulum bob has kinetic energy and no potential energy (assuming we take the reference level where potential energy is zero at the starting height).
- The initial kinetic energy (KE) of the bob is given by:
\[ KE = \frac{1}{2} mv^2 \]
- The potential energy (PE) at the maximum height \( h \) is given by:
\[ PE = mgh \]

4. **Set Up the Energy Conservation Equation**:
- According to the conservation of energy:
\[ \text{Initial KE} = \text{Final PE} \]
- Therefore, we can write:
\[ \frac{1}{2} mv^2 = mgh \]

5. **Solve for Maximum Height \( h \)**:
- We can cancel the mass \( m \) from both sides of the equation (since \( m \) is not zero):
\[ \frac{1}{2} v^2 = gh \]
- Rearranging gives us:
\[ h = \frac{v^2}{2g} \]

6. **Conclusion**:
- The maximum height attained by the pendulum bob after the collision is:
\[ h = \frac{v^2}{2g} \]

### Final Answer:
The maximum height attained by the bob after the collision is \( \frac{v^2}{2g} \).

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