A 1.2 kg block is held at rest against the spring with a force constant k= 730 N/m. Initially, the spring is compressed a distance d. When the block is released, it slides across a surface that is frictionless except for a rough patch of width 5.0 cm that has a coefficient of kinetic friction = 0.44. Find d, such that the block's speed after crossing the rough patch is 2.3 m/s.

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:

the total distance that martina travels that day is

(distance from home to mom's office) plus (distance from mom's office to school)

plus (distance from school to library) plus (distance from library to friend's house)

plus (distance from friend's house to martina's house).

(we don't know what any of these distance actually is.)