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Kinetic energy is a crucial concept in physics that refers to the energy that an object possesses due to its motion. The formula to calculate kinetic energy (KE) is: \[ KE = \frac{1}{2} mv^2 \] where:

• \(m\) is the mass of the object in kilograms,
• \(v\) is the velocity of the object in meters per second.

In our exercise, the water-skier has both initial and final speeds. Therefore, we need to calculate both the initial kinetic energy \(KE_i\) and the final kinetic energy \(KE_f\). This calculation helps in understanding how much energy is gained or lost by the skier as her speed changes. By substituting the skier's mass and velocities into the kinetic energy formula, we first find the initial state kinetic energy. Then, we do the same for the final state to understand how the skier's energy has transformed. These are preliminary calculations before using the work-energy principle.

Work done is another foundational idea in physics, directly connected to energy changes. In this context, work done refers to the energy transferred by the force to cause the skier's acceleration. The work done by a force is given by the equation: \[ W = F \cdot d \cdot \cos(\theta) \] where:

• \(F\) is the force applied,
• \(d\) is the distance over which the force is applied,
• \(\theta\) is the angle between the force and displacement directions.

However, when using the work-energy principle, we focus on the change in kinetic energy to determine the work done. For the skier, this can be found using: \[ W = KE_f - KE_i \] This formula indicates that the work done on the skier by the net external force is essentially the difference in her kinetic energy from start to finish. It translates the forces acting upon her into a measurable energy change, which explains her acceleration.

Net external force is the key player in changing the state of motion of an object and is defined as the overall force resulting from all the external forces acting on an object. In physics, it's what causes acceleration, which is famously stated in Newton's second law: \[ F_{net} = ma \] where:

• \(F_{net}\) is the net external force,
• \(m\) is the mass of the object,
• \(a\) is the acceleration.

In the water-skier scenario, the net external force results from the boat's pull through the tow rope. This force is responsible for the skier's acceleration from an initial speed to a higher speed. By calculating the work done using the change in kinetic energy, we're effectively determining how much energy the net external force contributes to make the skier go faster. Understanding the net external force not only helps us compute work but also deepens our grasp of the dynamics involved in motion and energy transfer in real-world situations.