A point traversed half the distance with a velocity `v_0`. The remaining part of the distance was covered with velocity `v_1` for half the time, and with velocity `v_2` for the other half of the time. Find the mean velocity of the point averaged over the whole time of motion.

Explanation

To find the mean velocity of the point averaged over the whole time of motion, we can follow these steps:

### Step-by-Step Solution

1. **Define the total distance**:
Let the total distance be \( D \). The point travels half of this distance with velocity \( v_0 \) and the other half with velocities \( v_1 \) and \( v_2 \).

\[ \text{Total distance} = 2D \]

2. **Calculate the time taken for the first half of the distance**:
The time taken to cover the first half of the distance \( D \) at velocity \( v_0 \) can be calculated using the formula:

\[ T_1 = \frac{D}{v_0} \]

3. **Calculate the time taken for the second half of the distance**:
The second half of the distance \( D \) is covered in two parts: half the time at velocity \( v_1 \) and the other half at velocity \( v_2 \). Let the time for each part be \( T_2 \).

The total distance covered in the second half can be expressed as:

\[ v_1 T_2 + v_2 T_2 = D \]

This simplifies to:

\[ (v_1 + v_2) T_2 = D \]

Thus, we can find \( T_2 \):

\[ T_2 = \frac{D}{v_1 + v_2} \]

4. **Calculate the total time taken**:
The total time taken \( T \) for the entire journey is the sum of the times for both halves:

\[ T = T_1 + 2T_2 = \frac{D}{v_0} + 2 \left( \frac{D}{v_1 + v_2} \right) \]

This simplifies to:

\[ T = \frac{D}{v_0} + \frac{2D}{v_1 + v_2} \]

5. **Calculate the mean velocity**:
The mean velocity \( V_{avg} \) is defined as the total distance traveled divided by the total time taken:

\[ V_{avg} = \frac{\text{Total distance}}{\text{Total time}} = \frac{2D}{T} \]

Substituting for \( T \):

\[ V_{avg} = \frac{2D}{\frac{D}{v_0} + \frac{2D}{v_1 + v_2}} \]

Simplifying this expression:

\[ V_{avg} = \frac{2D}{D \left( \frac{1}{v_0} + \frac{2}{v_1 + v_2} \right)} = \frac{2}{\frac{1}{v_0} + \frac{2}{v_1 + v_2}} \]

Further simplifying gives:

\[ V_{avg} = \frac{2v_0(v_1 + v_2)}{v_1 + v_2 + 2v_0} \]

### Final Result
Thus, the mean velocity of the point averaged over the whole time of motion is:

\[ V_{avg} = \frac{2v_0(v_1 + v_2)}{v_1 + v_2 + 2v_0} \]