To tackle this problem, we need to break down the motion of the block into segments and analyze the average velocity over the entire journey. The average velocity is defined as the total displacement divided by the total time taken. Let's go through the steps together.
Understanding the Motion Segments
The block moves in three distinct segments:
- First segment: Moves with velocity v for time t.
- Second segment: Moves with velocity 2v for time t.
- Third segment: Moves with velocity 3v for time T1.
Calculating Displacement for Each Segment
Now, let's calculate the displacement for each segment:
- Displacement during the first segment: d1 = v * t
- Displacement during the second segment: d2 = 2v * t
- Displacement during the third segment: d3 = 3v * T1
Total Displacement
The total displacement D for the entire journey is the sum of the displacements from all three segments:
D = d1 + d2 + d3 = vt + 2vt + 3vT1 = vt(1 + 2) + 3vT1 = 3vt + 3vT1
Calculating Total Time
The total time T taken for the journey is the sum of the time intervals for each segment:
T = t + t + T1 = 2t + T1
Average Velocity Formula
The average velocity V_avg for the entire journey is given as:
V_avg = Total Displacement / Total Time
Substituting the values we calculated:
2.5v = (3vt + 3vT1) / (2t + T1)
Solving for T1
Now, let's simplify and solve for T1:
- Multiply both sides by (2t + T1):
- 2.5v(2t + T1) = 3vt + 3vT1
- Distributing the left side gives:
- 5vt + 2.5vT1 = 3vt + 3vT1
- Rearranging terms leads to:
- 5vt - 3vt = 3vT1 - 2.5vT1
- This simplifies to:
- 2vt = 0.5vT1
- Dividing both sides by v (assuming v ≠ 0):
- 2t = 0.5T1
- Multiplying both sides by 2 gives:
- T1 = 4t
Final Thoughts
So, the value of T1 in terms of t is indeed 4t. This means that during the last segment of the journey, the block moves for four times the duration of the first two segments combined. If you have any further questions or need clarification on any part of this, feel free to ask!
