Two masses of 0.25 kg each moves toward each other with speed 3 `ms^(-1)` collide and stick together. Find the final velocity

Answer

To solve the problem of two masses colliding and sticking together, we can use the principle of conservation of linear momentum. Here’s a step-by-step solution:

### Step 1: Identify the masses and their velocities
Let:
- Mass of body 1 (m1) = 0.25 kg
- Mass of body 2 (m2) = 0.25 kg
- Velocity of body 1 (v1) = 3 m/s (moving to the right)
- Velocity of body 2 (v2) = -3 m/s (moving to the left, hence negative)

### Step 2: Write the equation for conservation of momentum
According to the conservation of momentum:
\[ \text{Initial Momentum} = \text{Final Momentum} \]

The initial momentum (p_initial) can be calculated as:
\[ p_{\text{initial}} = m_1 \cdot v_1 + m_2 \cdot v_2 \]

Substituting the values:
\[ p_{\text{initial}} = (0.25 \, \text{kg}) \cdot (3 \, \text{m/s}) + (0.25 \, \text{kg}) \cdot (-3 \, \text{m/s}) \]

### Step 3: Calculate the initial momentum
Calculating the individual terms:
\[ p_{\text{initial}} = 0.75 \, \text{kg m/s} - 0.75 \, \text{kg m/s} \]
\[ p_{\text{initial}} = 0 \, \text{kg m/s} \]

### Step 4: Set up the equation for final momentum
After the collision, the two masses stick together, so the total mass after the collision is:
\[ m_{\text{total}} = m_1 + m_2 = 0.25 \, \text{kg} + 0.25 \, \text{kg} = 0.5 \, \text{kg} \]

Let the final velocity after the collision be \( v_f \). The final momentum (p_final) is:
\[ p_{\text{final}} = m_{\text{total}} \cdot v_f = 0.5 \, \text{kg} \cdot v_f \]

### Step 5: Equate initial and final momentum
Setting the initial momentum equal to the final momentum:
\[ 0 = 0.5 \, \text{kg} \cdot v_f \]

### Step 6: Solve for final velocity
From the equation:
\[ 0.5 \, \text{kg} \cdot v_f = 0 \]
This implies:
\[ v_f = 0 \, \text{m/s} \]

### Conclusion
The final velocity of the two masses after they collide and stick together is **0 m/s**.

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