A ball of mass 'm' moving with a horizontal velocity 'v' strikes the bob of mass 'm' of a pendulum at rest. During this collision, the ball sticks with the bob of the pendulum. The height to which the combined mass raises is (g = acceleration due to gravity).

Options

A. `(v^(2))/(4 g)`
B. `(v^(2))/(8g)`
C. `(v^(2))/(g)`
D. `(v^(2))/(2g)`

Explanation

To solve the problem, we need to analyze the collision between the ball and the pendulum bob, and then determine how high the combined mass rises after the collision. Here’s a step-by-step solution:

### Step 1: Understand the Collision
We have a ball of mass \( m \) moving with a horizontal velocity \( v \) that strikes a pendulum bob of mass \( m \) at rest. The collision is perfectly inelastic, meaning the ball sticks to the bob after the collision.

### Step 2: Apply Conservation of Momentum
Before the collision, the momentum of the system is solely due to the moving ball:
\[ \text{Initial Momentum} = mv \]
After the collision, both the ball and the bob move together with a common velocity \( v' \). The total mass after the collision is \( 2m \). Therefore, the momentum after the collision can be expressed as:
\[ \text{Final Momentum} = (2m)v' \]
By the conservation of momentum:
\[ mv = (2m)v' \]
Cancelling \( m \) from both sides (assuming \( m \neq 0 \)):
\[ v = 2v' \quad \Rightarrow \quad v' = \frac{v}{2} \]

### Step 3: Analyze the Motion After the Collision
After the collision, the combined mass (ball + bob) will rise to a certain height \( h \). We can use the conservation of energy principle to find this height. The kinetic energy just after the collision will be converted into potential energy at the maximum height.

### Step 4: Write the Energy Conservation Equation
The kinetic energy (KE) just after the collision is:
\[ \text{KE} = \frac{1}{2}(2m)(v')^2 = m(v')^2 \]
Substituting \( v' = \frac{v}{2} \):
\[ \text{KE} = m\left(\frac{v}{2}\right)^2 = m\frac{v^2}{4} = \frac{mv^2}{4} \]

The potential energy (PE) at height \( h \) is given by:
\[ \text{PE} = (2m)gh \]
Setting the kinetic energy equal to the potential energy:
\[ \frac{mv^2}{4} = (2m)gh \]

### Step 5: Solve for Height \( h \)
Now we can simplify the equation:
\[ \frac{mv^2}{4} = 2mgh \]
Dividing both sides by \( 2m \) (again assuming \( m \neq 0 \)):
\[ \frac{v^2}{8} = gh \]
Rearranging for \( h \):
\[ h = \frac{v^2}{8g} \]

### Final Answer
The height to which the combined mass raises is:
\[ h = \frac{v^2}{8g} \]