A block of mass 1 kg, moving along x with speed 
enters a rough region ranging from 
to 
. The retarding force acting on the block in this range is 
, with 
. Then the final speed of the block as it crosses rough region is.
Text Solution
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To find the final speed of a block moving through a rough region, consider the following steps:
Given Parameters:
Initial speed of the block, 
.
Mass of the block, 
.
Force of friction acting on the block, 
.
Forces and Acceleration:
The acceleration 
of the block is given by 
.
So, we have: 
.
Using the Relationship:
We use the relationship between velocity, acceleration, and position: 
.
Substituting 
gives: 
.
Integrating the Equation:
Integrate both sides with respect to their variables:
Solving the Integrals:
The left side becomes 
.
The right side evaluates to 
.
Therefore:
Final Calculation:
Solve the above equation for 
:
Simplifying and solving gives:
Solving further:
Therefore, the final speed of the block as it exits the rough region is approximately 
.
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