A particle moves along x-axis with speed `6 m//s` for the first half distance of a journey and the second half distance with a speed `3 m//s`. The average speed in the total journey is

Options

A. `5 m//s`
B. `4.5 m//s`
C. `4 m//s`
D. `2 m//s`

Answer

To find the average speed of a particle moving along the x-axis with different speeds for two halves of the journey, we can follow these steps:

### Step 1: Define the total distance
Let the total distance of the journey be \( x \). Therefore, the first half distance is \( \frac{x}{2} \) and the second half distance is also \( \frac{x}{2} \).

### Step 2: Calculate time for the first half of the journey
The speed for the first half of the journey is \( 6 \, \text{m/s} \). Using the formula for time, \( \text{time} = \frac{\text{distance}}{\text{speed}} \):
\[ t_1 = \frac{\frac{x}{2}}{6} = \frac{x}{12} \, \text{seconds} \]

### Step 3: Calculate time for the second half of the journey
The speed for the second half of the journey is \( 3 \, \text{m/s} \). Again using the time formula:
\[ t_2 = \frac{\frac{x}{2}}{3} = \frac{x}{6} \, \text{seconds} \]

### Step 4: Calculate total time for the journey
The total time \( T \) taken for the journey is the sum of the times for both halves:
\[ T = t_1 + t_2 = \frac{x}{12} + \frac{x}{6} \]
To add these fractions, find a common denominator (which is 12):
\[ T = \frac{x}{12} + \frac{2x}{12} = \frac{3x}{12} = \frac{x}{4} \, \text{seconds} \]

### Step 5: Calculate average speed
The average speed \( V_{avg} \) is given by the formula:
\[ V_{avg} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{x}{T} \]
Substituting the values we found:
\[ V_{avg} = \frac{x}{\frac{x}{4}} = 4 \, \text{m/s} \]

### Conclusion
The average speed for the total journey is \( 4 \, \text{m/s} \).

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