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 times with different speeds, we can follow these steps:

### Step 1: Calculate the distance traveled in the first segment
The person travels for the first \( \frac{t}{3} \) time with a speed \( v_1 \). The distance \( d_1 \) traveled during this time can be calculated using the formula:
\[ d_1 = v_1 \times \text{time} = v_1 \times \frac{t}{3} \]

### Step 2: Calculate the distance traveled in the second segment
For the next \( \frac{2t}{3} \) time, the person travels with a speed \( v_2 \). The distance \( d_2 \) for this segment can be calculated as:
\[ d_2 = v_2 \times \text{time} = v_2 \times \frac{2t}{3} \]

### Step 3: Calculate the total distance traveled
The total distance \( d \) traveled by the person is the sum of the distances from both segments:
\[ d = d_1 + d_2 = v_1 \times \frac{t}{3} + v_2 \times \frac{2t}{3} \]

### Step 4: Calculate the total time taken
The total time \( T \) taken for the journey is the sum of the times for both segments:
\[ T = \frac{t}{3} + \frac{2t}{3} = t \]

### Step 5: Calculate the mean speed
The mean speed \( v \) is defined as the total distance traveled divided by the total time taken:
\[ v = \frac{d}{T} = \frac{v_1 \times \frac{t}{3} + v_2 \times \frac{2t}{3}}{t} \]

### Step 6: Simplify the expression
Substituting the total distance and total time into the mean speed formula gives:
\[ v = \frac{v_1 \times \frac{t}{3} + v_2 \times \frac{2t}{3}}{t} \]
We can factor out \( \frac{t}{3} \) from the numerator:
\[ v = \frac{\frac{t}{3} (v_1 + 2v_2)}{t} \]
Now, the \( t \) in the numerator and denominator cancels out:
\[ v = \frac{v_1 + 2v_2}{3} \]

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