A body moves speed `V_1` for distance L and then with speed `V_2` for distance 2L. The average speed for the motion is

Options

A. `(V_1 + V_2)/(2)`
B. `(3V_1 V_2)/(V_1 +2 V_2)`
C. `(3V_1 V_2)/(2V_1 + V_2)`
D. `(3V_1 V_2)/(V_1 + V_2)`

Answer

To find the average speed of a body moving at different speeds over different distances, we can follow these steps:

### Step 1: Identify the distances and speeds
The body moves:
- Distance \( L \) at speed \( V_1 \)
- Distance \( 2L \) at speed \( V_2 \)

### Step 2: Calculate the total distance
The total distance \( D \) traveled by the body is the sum of the two distances:
\[ D = L + 2L = 3L \]

### Step 3: Calculate the time taken for each segment
Using the formula for time, \( t = \frac{d}{v} \):
- For the first segment (distance \( L \) at speed \( V_1 \)):
\[ T_1 = \frac{L}{V_1} \]
- For the second segment (distance \( 2L \) at speed \( V_2 \)):
\[ T_2 = \frac{2L}{V_2} \]

### Step 4: Calculate the total time
The total time \( T \) taken for the entire journey is the sum of the times for both segments:
\[ T = T_1 + T_2 = \frac{L}{V_1} + \frac{2L}{V_2} \]

### Step 5: Find a common denominator for the total time
To combine the fractions, we can find a common denominator:
\[ T = \frac{L V_2}{V_1 V_2} + \frac{2L V_1}{V_1 V_2} = \frac{L V_2 + 2L V_1}{V_1 V_2} = \frac{L(V_2 + 2V_1)}{V_1 V_2} \]

### Step 6: Calculate the average speed
The average speed \( V_{avg} \) is defined as the total distance divided by the total time:
\[ V_{avg} = \frac{D}{T} = \frac{3L}{\frac{L(V_2 + 2V_1)}{V_1 V_2}} = \frac{3L \cdot V_1 V_2}{L(V_2 + 2V_1)} \]
The \( L \) cancels out:
\[ V_{avg} = \frac{3 V_1 V_2}{V_2 + 2V_1} \]

### Final Answer
The average speed for the motion is:
\[ V_{avg} = \frac{3 V_1 V_2}{V_2 + 2V_1} \]