A person travelling on a straight line moves with a uniform velocity `v_1` for a distance x and with a uniform velocity `v_2` for the next equal distance. The average velocity v is given by

Options

A. `v=(v_1+v_2)/2`
B. `v=sqrt(v_1v_2)`
C. `2/v=1/v_1+1/v_2`
D. `1/v=1/v_1+1/v_2`

Answer

To solve the problem step by step, we will calculate the average velocity of a person traveling with two different uniform velocities over equal distances.

### Step 1: Understand the problem
The person travels a distance \( x \) with a velocity \( v_1 \) and then travels another distance \( x \) with a velocity \( v_2 \). We need to find the average velocity \( v \) for the entire journey.

### Step 2: Calculate total displacement
The total displacement \( D \) is the sum of the two distances traveled:
\[ D = x + x = 2x \]

### Step 3: Calculate total time taken
To find the total time \( T \), we will calculate the time taken for each segment of the journey.

- Time taken to travel the first distance \( x \) with velocity \( v_1 \):
\[ T_1 = \frac{x}{v_1} \]

- Time taken to travel the second distance \( x \) with velocity \( v_2 \):
\[ T_2 = \frac{x}{v_2} \]

Now, we can find the total time \( T \):
\[ T = T_1 + T_2 = \frac{x}{v_1} + \frac{x}{v_2} \]

### Step 4: Simplify the total time
Factor out \( x \) from the total time equation:
\[ T = x \left( \frac{1}{v_1} + \frac{1}{v_2} \right) \]

### Step 5: Calculate average velocity
The average velocity \( v \) is given by the formula:
\[ v = \frac{\text{Total Displacement}}{\text{Total Time}} = \frac{D}{T} \]
Substituting the values we found:
\[ v = \frac{2x}{x \left( \frac{1}{v_1} + \frac{1}{v_2} \right)} \]

### Step 6: Cancel \( x \)
Since \( x \) appears in both the numerator and denominator, we can cancel it out:
\[ v = \frac{2}{\frac{1}{v_1} + \frac{1}{v_2}} \]

### Step 7: Final expression for average velocity
To express this in a more standard form, we can take the reciprocal:
\[ \frac{1}{v} = \frac{1}{2} \left( \frac{1}{v_1} + \frac{1}{v_2} \right) \]
Thus, we can express the average velocity as:
\[ v = \frac{2v_1v_2}{v_1 + v_2} \]

### Final Answer
The average velocity \( v \) is given by:
\[ v = \frac{2v_1v_2}{v_1 + v_2} \]

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