A 50 g bullet moving with velocity 10 m / s strikes a block of mass 950 g at rest and gets embedded in it. The loss in kinetic energy will be

Options

A. 1
B. 2.375
C. 0.05
D. 0.5

Answer

To solve the problem of finding the loss in kinetic energy when a bullet embeds itself in a block, we can follow these steps:

### Step 1: Convert the masses into kilograms
The mass of the bullet is given as 50 g, and the mass of the block is 950 g. We need to convert these masses into kilograms for consistency in SI units.

- Mass of bullet, \( m_b = 50 \, \text{g} = 0.050 \, \text{kg} \)
- Mass of block, \( m_B = 950 \, \text{g} = 0.950 \, \text{kg} \)

### Step 2: Calculate the initial kinetic energy of the bullet
The initial kinetic energy (KE) of the bullet can be calculated using the formula:
\[ KE = \frac{1}{2} mv^2 \]
where \( m \) is the mass and \( v \) is the velocity.

- Velocity of bullet, \( v_b = 10 \, \text{m/s} \)

Calculating the initial kinetic energy of the bullet:
\[ KE_{\text{initial}} = \frac{1}{2} \times 0.050 \, \text{kg} \times (10 \, \text{m/s})^2 \]
\[ KE_{\text{initial}} = \frac{1}{2} \times 0.050 \times 100 = 2.5 \, \text{J} \]

### Step 3: Calculate the final velocity after the collision
Since the bullet embeds itself in the block, we can use the conservation of momentum to find the final velocity \( v_f \) after the collision.

The initial momentum of the system (only the bullet is moving):
\[ \text{Initial Momentum} = m_b \cdot v_b + m_B \cdot 0 = 0.050 \cdot 10 + 0.950 \cdot 0 = 0.5 \, \text{kg m/s} \]

The total mass after the collision:
\[ m_{\text{total}} = m_b + m_B = 0.050 + 0.950 = 1.000 \, \text{kg} \]

Using conservation of momentum:
\[ \text{Initial Momentum} = \text{Final Momentum} \]
\[ 0.5 = 1.000 \cdot v_f \]
\[ v_f = 0.5 \, \text{m/s} \]

### Step 4: Calculate the final kinetic energy of the combined system
Now we can calculate the final kinetic energy of the system (bullet + block) using the final velocity \( v_f \):
\[ KE_{\text{final}} = \frac{1}{2} m_{\text{total}} v_f^2 \]
\[ KE_{\text{final}} = \frac{1}{2} \times 1.000 \, \text{kg} \times (0.5 \, \text{m/s})^2 \]
\[ KE_{\text{final}} = \frac{1}{2} \times 1.000 \times 0.25 = 0.125 \, \text{J} \]

### Step 5: Calculate the loss in kinetic energy
The loss in kinetic energy can be found by subtracting the final kinetic energy from the initial kinetic energy:
\[ \text{Loss in KE} = KE_{\text{initial}} - KE_{\text{final}} \]
\[ \text{Loss in KE} = 2.5 \, \text{J} - 0.125 \, \text{J} = 2.375 \, \text{J} \]

### Final Answer
The loss in kinetic energy is \( 2.375 \, \text{J} \).

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