On an essentially frictionless, horizontal ice rink, a skater moving at 5.0 m/s encounters a rough patch that reduces her speed by 50 % due to a friction force that is 22 % of her weight. Use the work–energy theorem to find the length of this rough patch

The length of the rough patch is 4.345 meters.

Explanation:

According to the Work-Energy Theorem, change in kinetic energy is equal to the dissipated work due to friction. That is:

Where:

, - Initial and final kinetic energy, measured in joules.

- Work losses due to friction.

By applying definitions of kinetic energy and work, the expression described above is expanded:

Where:

, - Initial and final speed of the skater, measured in meters per second.

- Mass of the skater, measured in kilograms.

- Length of the rough patch, measured in meters.

- Friction force, measured in newtons.

According to the statement, friction force is represented by the following expression:

Where:

- Ratio of friction force to weight, dimensionless.

- Gravitational constant, measured in meters per square second.

Then,

The equation is simplified algebraically and patch length is cleared afterwards:

Given that , , and , the length of the rough patch is:

The length of the rough patch is 4.345 meters.

not too sure thinking all accelerations are pointed down