A block of mass 0.50 kg is moving with a speed of 2.00 m/s on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is

Options

A. 0.16 J
B. 1.00 J
C. 0.67 J
D. 0.34 J

Explanation

To solve the problem of energy loss during the collision, we will follow these steps:

### Step 1: Identify the masses and initial velocities
- Mass of block 1 (m1) = 0.50 kg
- Initial velocity of block 1 (v1) = 2.00 m/s
- Mass of block 2 (m2) = 1.00 kg
- Initial velocity of block 2 (v2) = 0 m/s (at rest)

### Step 2: Calculate the initial kinetic energy
The initial kinetic energy (KE_initial) of the system can be calculated using the formula:
\[ KE_{initial} = \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 \]
Substituting the values:
\[ KE_{initial} = \frac{1}{2} (0.50) (2.00)^2 + \frac{1}{2} (1.00) (0)^2 \]
\[ KE_{initial} = \frac{1}{2} (0.50) (4) + 0 = 1.00 \text{ J} \]

### Step 3: Apply conservation of momentum to find the final velocity
Since the two blocks stick together after the collision, we can use the conservation of momentum:
\[ m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_f \]
Substituting the known values:
\[ (0.50)(2.00) + (1.00)(0) = (0.50 + 1.00) v_f \]
\[ 1.00 = 1.50 v_f \]
Solving for \(v_f\):
\[ v_f = \frac{1.00}{1.50} = \frac{2}{3} \text{ m/s} \]

### Step 4: Calculate the final kinetic energy
Now, we can calculate the final kinetic energy (KE_final) of the combined mass:
\[ KE_{final} = \frac{1}{2} (m_1 + m_2) v_f^2 \]
Substituting the values:
\[ KE_{final} = \frac{1}{2} (1.50) \left(\frac{2}{3}\right)^2 \]
Calculating:
\[ KE_{final} = \frac{1}{2} (1.50) \left(\frac{4}{9}\right) = \frac{1.50 \times 4}{18} = \frac{6}{18} = \frac{1}{3} \text{ J} \]

### Step 5: Calculate the energy loss
The energy loss during the collision can be found by subtracting the final kinetic energy from the initial kinetic energy:
\[ \text{Energy loss} = KE_{initial} - KE_{final} \]
Substituting the values:
\[ \text{Energy loss} = 1.00 \text{ J} - \frac{1}{3} \text{ J} = \frac{3}{3} - \frac{1}{3} = \frac{2}{3} \text{ J} \]

### Final Answer
The energy loss during the collision is \(\frac{2}{3} \text{ J}\) or approximately \(0.67 \text{ J}\).

---