Four particles have velocities `1, 0,2, and 3 m//s`. The root mean square velocity of the particles (definition wise) is.

Options

A. `3.5 m//s`.
B. `sqrt (3.5 m//s)`.
C. `1.5 m//s`
D. `sqrt((14)/(3)) m//s`.

Explanation

To find the root mean square (RMS) velocity of the four particles with velocities 1 m/s, 0 m/s, 2 m/s, and 3 m/s, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the velocities of the particles:**
- The velocities are given as \( v_1 = 1 \, \text{m/s} \), \( v_2 = 0 \, \text{m/s} \), \( v_3 = 2 \, \text{m/s} \), and \( v_4 = 3 \, \text{m/s} \).

2. **Use the formula for root mean square velocity:**
- The formula for root mean square velocity (RMS) is given by:
\[ v_{\text{rms}} = \sqrt{\frac{v_1^2 + v_2^2 + v_3^2 + v_4^2}{n}} \]
where \( n \) is the number of particles.

3. **Calculate the squares of the velocities:**
- Calculate \( v_1^2 = 1^2 = 1 \)
- Calculate \( v_2^2 = 0^2 = 0 \)
- Calculate \( v_3^2 = 2^2 = 4 \)
- Calculate \( v_4^2 = 3^2 = 9 \)

4. **Sum the squares of the velocities:**
- Add the squares:
\[ v_1^2 + v_2^2 + v_3^2 + v_4^2 = 1 + 0 + 4 + 9 = 14 \]

5. **Divide by the number of particles:**
- Since there are 4 particles, we divide the sum by 4:
\[ \frac{14}{4} = 3.5 \]

6. **Take the square root:**
- Finally, take the square root of the result:
\[ v_{\text{rms}} = \sqrt{3.5} \]

7. **Calculate the final value:**
- The approximate value of \( \sqrt{3.5} \) is about \( 1.87 \, \text{m/s} \).

### Final Answer:
The root mean square velocity of the particles is approximately \( 1.87 \, \text{m/s} \).

---