To solve this problem, we need to analyze the motion of both the boy and the block, taking into account their respective accelerations and the relationship between their movements due to the pulley system. Let's break it down step by step.

Understanding the System

We have a boy and a block, both with the same mass, connected by a string over a frictionless pulley. When the boy accelerates upwards, the block will move downwards due to the tension in the string. The boy's acceleration is given as 2.5 m/s² relative to the rope.

Acceleration of the Block

Since the boy is moving upwards with an acceleration of 2.5 m/s² relative to the rope, we need to determine the actual acceleration of the block. In a pulley system like this, the acceleration of the block is equal to the acceleration of the boy, but in the opposite direction. Therefore, if the boy accelerates upwards, the block accelerates downwards with the same magnitude of acceleration.

However, we need to consider the relative motion. The boy's upward acceleration of 2.5 m/s² means that the block will experience a downward acceleration of 2.5 m/s² as well. Thus, the block's acceleration is:

• Acceleration of the block (a): 2.5 m/s² (downwards)

Calculating Time Taken by the Block

Next, we need to find out how long it takes for the block to travel a distance of 10 m. We can use the second equation of motion, which relates distance (s), initial velocity (u), acceleration (a), and time (t):

s = ut + (1/2)at²

In this scenario:

• s: 10 m (the distance the block travels)
• u: 0 m/s (the block starts from rest)
• a: 2.5 m/s² (the acceleration of the block)

Substituting these values into the equation gives us:

10 = 0*t + (1/2)(2.5)t²

This simplifies to:

10 = 1.25t²

Now, we can solve for t²:

t² = 10 / 1.25

t² = 8

Taking the square root of both sides gives us:

t = √8 = 2.83 seconds (approximately)

Final Thoughts

Thus, the time taken by the block to travel a total distance of 10 m before reaching the pulley is approximately 2.83 seconds. This analysis illustrates how the acceleration of one object in a pulley system directly affects the motion of the other object, allowing us to calculate the time taken for a specific distance traveled.