A body of mass 0.1 kg travelling with a velocity of 5 m/s, collides head-on with another body of mass 0.2 kg travelling in the same direction with a velocity of 1.2 m/s. If the two bodies stick together after collision, their common velocity is

Options

A. `0.967" m"//s"`
B. `1.233" m"//"s"`
C. `2.467" m"//"s"`
D. `2.524" m"//"s"`

Answer

To solve the problem of finding the common velocity after a head-on collision between two bodies, we can use the principle of conservation of momentum. Here’s a step-by-step solution:

### Step 1: Identify the masses and velocities
- Mass of the first body (m1) = 0.1 kg
- Velocity of the first body (u1) = 5 m/s
- Mass of the second body (m2) = 0.2 kg
- Velocity of the second body (u2) = 1.2 m/s

### Step 2: Calculate the initial momentum
The initial momentum (p_initial) of the system can be calculated using the formula:
\[ p_{\text{initial}} = m_1 \cdot u_1 + m_2 \cdot u_2 \]

Substituting the values:
\[ p_{\text{initial}} = (0.1 \, \text{kg} \cdot 5 \, \text{m/s}) + (0.2 \, \text{kg} \cdot 1.2 \, \text{m/s}) \]
\[ p_{\text{initial}} = 0.5 \, \text{kg m/s} + 0.24 \, \text{kg m/s} \]
\[ p_{\text{initial}} = 0.74 \, \text{kg m/s} \]

### Step 3: Calculate the total mass after collision
After the collision, the two bodies stick together, so the total mass (M) is:
\[ M = m_1 + m_2 \]
\[ M = 0.1 \, \text{kg} + 0.2 \, \text{kg} = 0.3 \, \text{kg} \]

### Step 4: Apply conservation of momentum
According to the conservation of momentum:
\[ p_{\text{initial}} = p_{\text{final}} \]
The final momentum (p_final) can be expressed as:
\[ p_{\text{final}} = M \cdot V \]
Where V is the common velocity after the collision.

Setting the initial momentum equal to the final momentum:
\[ 0.74 \, \text{kg m/s} = 0.3 \, \text{kg} \cdot V \]

### Step 5: Solve for the common velocity (V)
Now, we can solve for V:
\[ V = \frac{0.74 \, \text{kg m/s}}{0.3 \, \text{kg}} \]
\[ V = \frac{0.74}{0.3} \]
\[ V = 2.4667 \, \text{m/s} \]

Rounding to three decimal places, we get:
\[ V \approx 2.467 \, \text{m/s} \]

### Final Answer
The common velocity after the collision is approximately **2.467 m/s**.

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