91Ó°ÊÓ

In physics, work done by nonconservative forces involves energy transfer that is not recoverable as mechanical energy. Such forces include friction, air resistance, and applied forces like pushes or pulls. In this exercise, the skater does +3420 J of work against the ground. This positive work indicates energy input by the skater pushing himself forward. Meanwhile, friction, a nonconservative force, counters this movement by doing -715 J of work against him. These energies cause changes in both kinetic and potential energies of the system.
The work-energy principle equation, \( W_{\text{nc}} = \Delta KE + \Delta PE \), shows how nonconservative work contributes to energy changes in a system. It demonstrates that the sum of nonconservative work influences both kinetic (motion-related) and potential (position-related) energy. In this scenario, the skater’s efforts and friction's resistance affect his total energy balance.

Kinetic energy is the energy associated with the motion of an object. It is expressed mathematically as \( KE = \frac{1}{2}mv^2 \), where \( m \) is mass and \( v \) is velocity. Changes in kinetic energy are directly tied to changes in the object's speed.
In the given exercise, the skater’s initial speed is \( 2.50 \, \text{m/s} \) and his final speed is \( 1.22 \, \text{m/s} \). Calculating the change in kinetic energy using:

• \( \Delta KE = \frac{1}{2}m(v_f^2 - v_i^2) \)

This formula accounts for the reduction in speed, resulting in a drop in kinetic energy. The skater loses 192.8 J of kinetic energy, reflecting the energy consumed in overcoming friction and the gravitational potential to move uphill.

Potential energy is energy stored by an object's position, typically concerning height in gravitational fields. It is calculated with \( PE = mgh \), where \( m \) is mass, \( g \) is the gravitational constant \( 9.81 \, \text{m/s}^2 \), and \( h \) is height change.
The positive change in potential energy, 2897.8 J in this exercise, suggests the skater moved to a higher elevation. This is because potential energy increases as an object ascends. Using the formula for potential energy change, \( \Delta PE = mgh \), we solve for the change in height \( h \) to find the skater ascends approximately 3.66 meters. This uphill movement ties into the work-energy principle, showing how energy input by the skater raises his potential energy as he climbs against gravity.