Answer:
Compression distance: 
Explanation:
According to this statement, we know that system is non-conservative due to the rough patch. By Principle of Energy Conservation and Work-Energy Theorem, we have the following expression that represents the system having a translational kinetic energy (
), in joules, at the expense of elastic potential energy (
), in joules, and overcoming work losses due to friction (
), in joules:
(1)
By definitions of translational kinetic and elastic potential energies and work losses due to friction, we expand the equation described above:
(2)
Where:
- Mass of the block, in kilograms.
- Final velocity of the block, in meters per second.
- KInetic coefficient of friction, no unit.
- Gravitational acceleration, in meters per square second.
- Width of the rough patch, in meters.
- Spring constant, in newtons per meter.
- Compression distance, in meters.
If we know that 
, 
, 
, 
, 
and 
, then the compression distance of the spring is:
