A person travels along a straight road for the first `(t)/(3)` time with a speed `v_(1)` and for next `(2t)/(3)` time with a speed `v_(2)`. Then the mean speed v is given by

Options

A. `v=(v_(1)+2v_(2))/(3)`
B. `(1)/(v)=(1)/(3v_(1))+(2)/(3v_(2))`
C. `v=(1)/(3)sqrt(2v_(1)v_(2))`
D. `v=sqrt((3v_(2))/(2v_(1)))`

Answer

To find the mean speed \( v \) of a person who travels along a straight road for different time intervals and speeds, we can follow these steps:

### Step 1: Understand the problem
The person travels for the first \( \frac{t}{3} \) time with speed \( v_1 \) and for the next \( \frac{2t}{3} \) time with speed \( v_2 \). We need to find the mean speed over the total time.

### Step 2: Calculate the distance for each segment
1. **Distance for the first segment**:
- Time = \( \frac{t}{3} \)
- Speed = \( v_1 \)
- Distance \( D_1 = v_1 \times \text{time} = v_1 \times \frac{t}{3} = \frac{v_1 t}{3} \)

2. **Distance for the second segment**:
- Time = \( \frac{2t}{3} \)
- Speed = \( v_2 \)
- Distance \( D_2 = v_2 \times \text{time} = v_2 \times \frac{2t}{3} = \frac{2v_2 t}{3} \)

### Step 3: Calculate the total distance
- Total distance \( D \) is the sum of \( D_1 \) and \( D_2 \):
\[ D = D_1 + D_2 = \frac{v_1 t}{3} + \frac{2v_2 t}{3} = \frac{(v_1 + 2v_2)t}{3} \]

### Step 4: Calculate the total time
- Total time \( T \) is the sum of the two time intervals:
\[ T = \frac{t}{3} + \frac{2t}{3} = t \]

### Step 5: Calculate the mean speed
- Mean speed \( v \) is given by the formula:
\[ v = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{D}{T} \]
- Substituting the values we calculated:
\[ v = \frac{\frac{(v_1 + 2v_2)t}{3}}{t} = \frac{(v_1 + 2v_2)}{3} \]

### Final Result
Thus, the mean speed \( v \) is:
\[ v = \frac{v_1 + 2v_2}{3} \]

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