A particle of mass `m` moving with velocity `V_(0)` strikes a simple pendulum of mass `m` and sticks to it. The maximum height attained by the pendulum will be

Options

A. `(V_(0)^(2))/(8g)`
B. `sqrt(V_(0)g)`
C. `2 sqrt((V_(0))/(g))`
D. `V_(0)^(2)/(4g)`

Explanation

To solve the problem of a particle of mass `m` moving with velocity `V0` striking a simple pendulum of mass `m` and sticking to it, we can follow these steps:

### Step 1: Understand the system
Initially, we have a particle of mass `m` moving with velocity `V0`. When it strikes the pendulum (also of mass `m`), they stick together, forming a combined mass of `2m`.

### Step 2: Apply Conservation of Momentum
According to the conservation of momentum, the total momentum before the collision equals the total momentum after the collision.

- **Initial momentum (p_initial)**:
\[ p_{\text{initial}} = m \cdot V_0 \]

- **Final momentum (p_final)**:
After the collision, the combined mass moves with a new velocity `V`.
\[ p_{\text{final}} = (m + m) \cdot V = 2m \cdot V \]

Setting the initial momentum equal to the final momentum:
\[ m \cdot V_0 = 2m \cdot V \]

### Step 3: Solve for the final velocity (V)
We can simplify the equation by dividing both sides by `m` (assuming `m` is not zero):
\[ V_0 = 2V \implies V = \frac{V_0}{2} \]

### Step 4: Calculate the Kinetic Energy after the collision
The kinetic energy (KE) of the combined mass after the collision can be calculated using the formula:
\[ KE = \frac{1}{2} \cdot \text{mass} \cdot \text{velocity}^2 \]
Substituting the values:
\[ KE = \frac{1}{2} \cdot (2m) \cdot \left(\frac{V_0}{2}\right)^2 = \frac{1}{2} \cdot 2m \cdot \frac{V_0^2}{4} = \frac{m V_0^2}{4} \]

### Step 5: Set up the potential energy at maximum height
At the maximum height, all the kinetic energy will be converted into potential energy (PE). The potential energy at height `h` is given by:
\[ PE = \text{mass} \cdot g \cdot h \]
Substituting the mass of the combined system:
\[ PE = (2m) \cdot g \cdot h \]

### Step 6: Equate kinetic energy to potential energy
At the maximum height:
\[ KE = PE \]
Thus,
\[ \frac{m V_0^2}{4} = 2m \cdot g \cdot h \]

### Step 7: Solve for height (h)
Dividing both sides by `2m` (assuming `m` is not zero):
\[ \frac{V_0^2}{8} = g \cdot h \]
Now, solving for `h`:
\[ h = \frac{V_0^2}{8g} \]

### Final Answer
The maximum height attained by the pendulum is:
\[ h = \frac{V_0^2}{8g} \]