Consider an `1100` particels gas system with speeds distribution as follows :
1000 particles each with speed `100 m//s`
2000 particles each wityh speed `200 m//s`
4000 particles each with speed `300 m//s`
3000 particles each with speed `400 m//s` and 1000 particles each with speed `500 m//s`
Find the average speed, and rms speed.

Explanation

To solve the problem of finding the average speed and the root mean square (RMS) speed of a gas system with given particle speeds and their distributions, we can follow these steps:

### Step 1: Identify the given data
We have the following data for the gas particles:
- 1000 particles with speed \( V_1 = 100 \, \text{m/s} \)
- 2000 particles with speed \( V_2 = 200 \, \text{m/s} \)
- 4000 particles with speed \( V_3 = 300 \, \text{m/s} \)
- 3000 particles with speed \( V_4 = 400 \, \text{m/s} \)
- 1000 particles with speed \( V_5 = 500 \, \text{m/s} \)

### Step 2: Calculate the total number of particles
The total number of particles \( N \) is the sum of all particles:
\[ N = 1000 + 2000 + 4000 + 3000 + 1000 = 11000 \]

### Step 3: Calculate the average speed
The formula for average speed \( \bar{V} \) is given by:
\[ \bar{V} = \frac{N_1 V_1 + N_2 V_2 + N_3 V_3 + N_4 V_4 + N_5 V_5}{N} \]
Substituting the values:
\[ \bar{V} = \frac{1000 \times 100 + 2000 \times 200 + 4000 \times 300 + 3000 \times 400 + 1000 \times 500}{11000} \]
Calculating each term:
- \( 1000 \times 100 = 100000 \)
- \( 2000 \times 200 = 400000 \)
- \( 4000 \times 300 = 1200000 \)
- \( 3000 \times 400 = 1200000 \)
- \( 1000 \times 500 = 500000 \)

Adding these values:
\[ 100000 + 400000 + 1200000 + 1200000 + 500000 = 3200000 \]
Now, substituting back into the average speed formula:
\[ \bar{V} = \frac{3200000}{11000} \approx 290.91 \, \text{m/s} \]

### Step 4: Calculate the RMS speed
The formula for RMS speed \( V_{rms} \) is given by:
\[ V_{rms} = \sqrt{\frac{N_1 V_1^2 + N_2 V_2^2 + N_3 V_3^2 + N_4 V_4^2 + N_5 V_5^2}{N}} \]
Substituting the values:
\[ V_{rms} = \sqrt{\frac{1000 \times 100^2 + 2000 \times 200^2 + 4000 \times 300^2 + 3000 \times 400^2 + 1000 \times 500^2}{11000}} \]
Calculating each term:
- \( 1000 \times 100^2 = 10000000 \)
- \( 2000 \times 200^2 = 80000000 \)
- \( 4000 \times 300^2 = 360000000 \)
- \( 3000 \times 400^2 = 480000000 \)
- \( 1000 \times 500^2 = 250000000 \)

Adding these values:
\[ 10000000 + 80000000 + 360000000 + 480000000 + 250000000 = 1170000000 \]
Now, substituting back into the RMS speed formula:
\[ V_{rms} = \sqrt{\frac{1170000000}{11000}} \approx \sqrt{106363.6363} \approx 326.13 \, \text{m/s} \]

### Final Results
- Average Speed \( \bar{V} \approx 290.91 \, \text{m/s} \)
- RMS Speed \( V_{rms} \approx 326.13 \, \text{m/s} \)