A block moves in a straight line with velocity v for time `t_0.` Then, its velocity becomes 2v for next `t_0` time. Finally, its velocity becomes 3v for time T. If average velocity during the complete journey was 2. 5 v, then find T in terms of `t_0.`

Explanation

To solve the problem step by step, we will follow the logic presented in the video transcript.

### Step 1: Understand the problem
We need to find the time \( T \) in terms of \( t_0 \) given that a block moves with different velocities for specified durations and has an average velocity of \( 2.5v \) over the entire journey.

### Step 2: Calculate the displacements
1. **First segment**: The block moves with velocity \( v \) for time \( t_0 \).
\[ d_1 = v \cdot t_0 \]

2. **Second segment**: The block moves with velocity \( 2v \) for time \( t_0 \).
\[ d_2 = 2v \cdot t_0 \]

3. **Third segment**: The block moves with velocity \( 3v \) for time \( T \).
\[ d_3 = 3v \cdot T \]

### Step 3: Calculate total displacement
The total displacement \( D \) is the sum of the displacements from all three segments:
\[ D = d_1 + d_2 + d_3 = vt_0 + 2vt_0 + 3vT = 3vt_0 + 3vT \]

### Step 4: Calculate total time
The total time \( T_{total} \) is the sum of the times for all three segments:
\[ T_{total} = t_0 + t_0 + T = 2t_0 + T \]

### Step 5: Set up the average velocity equation
The average velocity \( V_{avg} \) is given by the formula:
\[ V_{avg} = \frac{D}{T_{total}} \]
We know that the average velocity is \( 2.5v \), so we can set up the equation:
\[ \frac{3vt_0 + 3vT}{2t_0 + T} = 2.5v \]

### Step 6: Simplify the equation
We can cancel \( v \) from both sides (assuming \( v \neq 0 \)):
\[ \frac{3t_0 + 3T}{2t_0 + T} = 2.5 \]

### Step 7: Cross-multiply to eliminate the fraction
\[ 3t_0 + 3T = 2.5(2t_0 + T) \]
Expanding the right side:
\[ 3t_0 + 3T = 5t_0 + 2.5T \]

### Step 8: Rearrange the equation
Bringing all terms involving \( T \) to one side and terms involving \( t_0 \) to the other side:
\[ 3T - 2.5T = 5t_0 - 3t_0 \]
\[ 0.5T = 2t_0 \]

### Step 9: Solve for \( T \)
Dividing both sides by \( 0.5 \):
\[ T = \frac{2t_0}{0.5} = 4t_0 \]

### Final Answer
Thus, the value of \( T \) in terms of \( t_0 \) is:
\[ T = 4t_0 \]