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 the particle over the entire journey, we can follow these steps:

### Step 1: Define the total distance
Let the total distance of the journey be \(2d\), where \(d\) is the distance for each half of the journey.

### Step 2: Calculate time for the first half of the journey
The speed for the first half of the journey is given as \(6 \, \text{m/s}\). The time taken to cover the first half distance \(d\) can be calculated using the formula:
\[ t_1 = \frac{d}{\text{speed}} = \frac{d}{6} \]

### Step 3: Calculate time for the second half of the journey
The speed for the second half of the journey is given as \(3 \, \text{m/s}\). The time taken to cover the second half distance \(d\) can be calculated as:
\[ t_2 = \frac{d}{\text{speed}} = \frac{d}{3} \]

### Step 4: Calculate total time taken for the journey
The total time taken for the entire journey is the sum of the time taken for both halves:
\[ t_{\text{total}} = t_1 + t_2 = \frac{d}{6} + \frac{d}{3} \]

To add these fractions, we need a common denominator. The common denominator for \(6\) and \(3\) is \(6\):
\[ t_{\text{total}} = \frac{d}{6} + \frac{2d}{6} = \frac{3d}{6} = \frac{d}{2} \]

### Step 5: Calculate average speed
The average speed \(v\) is defined as the total distance traveled divided by the total time taken:
\[ v = \frac{\text{total distance}}{\text{total time}} = \frac{2d}{t_{\text{total}}} = \frac{2d}{\frac{d}{2}} \]

### Step 6: Simplify the expression for average speed
Now, simplifying the expression:
\[ v = \frac{2d \cdot 2}{d} = 4 \, \text{m/s} \]

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

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