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 the body that moves at different speeds over different distances, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Distances and Speeds**:
- The body travels a distance \( L \) at speed \( V_1 \).
- Then, it travels a distance \( 2L \) at speed \( V_2 \).

2. **Calculate Total Distance**:
- The total distance \( D \) traveled by the body is:
\[ D = L + 2L = 3L \]

3. **Calculate Time Taken for Each Segment**:
- For the first segment (distance \( L \) at speed \( V_1 \)):
\[ T_1 = \frac{L}{V_1} \]
- For the second segment (distance \( 2L \) at speed \( V_2 \)):
\[ T_2 = \frac{2L}{V_2} \]

4. **Calculate Total Time**:
- The total time \( T \) taken for the entire journey is:
\[ T = T_1 + T_2 = \frac{L}{V_1} + \frac{2L}{V_2} \]

5. **Calculate Average Speed**:
- The average speed \( V_{avg} \) is defined as the total distance divided by the total time:
\[ V_{avg} = \frac{D}{T} = \frac{3L}{\frac{L}{V_1} + \frac{2L}{V_2}} \]

6. **Simplify the Expression**:
- Factor out \( L \) from the denominator:
\[ V_{avg} = \frac{3L}{L\left(\frac{1}{V_1} + \frac{2}{V_2}\right)} = \frac{3}{\frac{1}{V_1} + \frac{2}{V_2}} \]
- To combine the fractions in the denominator, find a common denominator:
\[ V_{avg} = \frac{3}{\frac{V_2 + 2V_1}{V_1 V_2}} = \frac{3V_1 V_2}{V_2 + 2V_1} \]

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