Three identical particles with velocities `v_0hati`, `-3v_0hatj` and `5v_0hatk` collide successively with each other in such a way that they form a single particle. The velocity vector of resultant particle is

Options

A. (a) `v_0/3(hati+hatj+hatk)`
B. (b) `v_0/3(hati-hatj+hatk)`
C. (c) `v_0/3(hati-3hatj+hatk)`
D. (d) `v_0/3(hati-3hatj+5hatk)`

Explanation

To find the velocity vector of the resultant particle formed by the collision of three identical particles with given velocities, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Mass and Velocities of the Particles:**
Each particle has the same mass, denoted as \( m \). The velocities of the three particles are:
- Particle 1: \( \vec{v_1} = v_0 \hat{i} \)
- Particle 2: \( \vec{v_2} = -3v_0 \hat{j} \)
- Particle 3: \( \vec{v_3} = 5v_0 \hat{k} \)

2. **Calculate the Initial Momentum:**
The total initial momentum \( \vec{P}_{initial} \) of the system can be calculated as the sum of the momenta of the individual particles:
\[ \vec{P}_{initial} = m \vec{v_1} + m \vec{v_2} + m \vec{v_3} \]
Substituting the velocities:
\[ \vec{P}_{initial} = m(v_0 \hat{i}) + m(-3v_0 \hat{j}) + m(5v_0 \hat{k}) \]
\[ \vec{P}_{initial} = mv_0 \hat{i} - 3mv_0 \hat{j} + 5mv_0 \hat{k} \]

3. **Express the Total Initial Momentum:**
Factoring out \( m \):
\[ \vec{P}_{initial} = m(v_0 \hat{i} - 3v_0 \hat{j} + 5v_0 \hat{k}) \]

4. **Final Momentum of the Resultant Particle:**
After the collision, the three particles combine to form a single particle with mass \( 3m \) and velocity \( \vec{v} \). Thus, the final momentum \( \vec{P}_{final} \) can be expressed as:
\[ \vec{P}_{final} = (3m) \vec{v} \]

5. **Apply Conservation of Momentum:**
According to the conservation of momentum:
\[ \vec{P}_{initial} = \vec{P}_{final} \]
Therefore:
\[ m(v_0 \hat{i} - 3v_0 \hat{j} + 5v_0 \hat{k}) = 3m \vec{v} \]

6. **Cancel the Mass \( m \):**
Since \( m \) is common in both sides, we can cancel it:
\[ v_0 \hat{i} - 3v_0 \hat{j} + 5v_0 \hat{k} = 3 \vec{v} \]

7. **Solve for the Velocity Vector \( \vec{v} \):**
Dividing both sides by 3:
\[ \vec{v} = \frac{1}{3}(v_0 \hat{i} - 3v_0 \hat{j} + 5v_0 \hat{k}) \]
\[ \vec{v} = \frac{v_0}{3} \hat{i} - v_0 \hat{j} + \frac{5v_0}{3} \hat{k} \]

8. **Final Result:**
The velocity vector of the resultant particle is:
\[ \vec{v} = \frac{v_0}{3} \hat{i} - v_0 \hat{j} + \frac{5v_0}{3} \hat{k} \]