Four particles have speed 2, 3, 4 and 5 cm//s respectively. Then their rms speed (cm/s) will be

Options

A. 27
B. `27/2`
C. `sqrt54`
D. `sqrt54/2`

Answer

To calculate the root mean square (RMS) speed of the four particles with speeds 2 cm/s, 3 cm/s, 4 cm/s, and 5 cm/s, we can follow these steps:

### Step 1: Square the speeds of each particle
We need to square each of the given speeds:
- \( v_1 = 2 \) cm/s, so \( v_1^2 = 2^2 = 4 \) cm²/s²
- \( v_2 = 3 \) cm/s, so \( v_2^2 = 3^2 = 9 \) cm²/s²
- \( v_3 = 4 \) cm/s, so \( v_3^2 = 4^2 = 16 \) cm²/s²
- \( v_4 = 5 \) cm/s, so \( v_4^2 = 5^2 = 25 \) cm²/s²

### Step 2: Sum the squared speeds
Now, we add these squared values together:
\[ v_1^2 + v_2^2 + v_3^2 + v_4^2 = 4 + 9 + 16 + 25 = 54 \text{ cm}^2/\text{s}^2 \]

### Step 3: Divide the sum by the number of particles
Next, we divide the total by the number of particles (which is 4):
\[ \text{Mean of squares} = \frac{54}{4} = 13.5 \text{ cm}^2/\text{s}^2 \]

### Step 4: Take the square root of the mean
Finally, we take the square root of the mean of the squares to find the RMS speed:
\[ v_{rms} = \sqrt{13.5} \text{ cm/s} \]

### Step 5: Calculate the final value
Calculating the square root gives us:
\[ v_{rms} \approx 3.674 \text{ cm/s} \]

Thus, the RMS speed of the four particles is approximately \( \sqrt{13.5} \) cm/s or about 3.67 cm/s.

### Final Answer:
The RMS speed of the particles is approximately \( 3.674 \) cm/s.

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