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Gravitational potential energy is a form of energy related to the position of an object in a gravitational field. It is the energy an object possesses due to its height above the ground. The formula to calculate gravitational potential energy (or its change) is \( \Delta U = mgh \), where:

• \(m\) stands for mass in kilograms (kg)
• \(g\) is the acceleration due to gravity, which is approximately \(9.8\, \text{m/s}^2\) on the surface of Earth
• \(h\) is the height in meters (m) that the object has been raised above its reference point

The potential energy increases as the object is raised higher. Here, as the skier ascends the mountain using the lift, their gravitational potential energy increases by an amount determined by the skier's mass, gravity, and the vertical height gained.

Physics problems like this one require a structured approach to apply the basic principles in a clear manner. Typically, solving a physics problem involves:

• Understanding and visualizing the problem
• Identifying the known and unknown quantities
• Choosing the correct formula or equations
• Substituting known values and solving for unknowns
• Ensuring the units are consistent and adjusting if necessary

In this exercise, we identified the skier’s mass, the gravitational acceleration, and the need to find the height gain using trigonometry. By accurately substituting these values into the formula, we calculated the change in gravitational potential energy. Remember, the solution process often involves breaking the problem into smaller steps, each of which is manageable and logical.

Trigonometry forms an essential part of solving physics problems, particularly when dealing with angles and distances. In this problem, the angle \(14.6^{\circ}\) between the lift and the horizontal is crucial to determining the vertical distance moved by the skier. To find the height, we use the sine (\(\sin\)) function, relating the hypotenuse (the length of the lift) and the opposite side (height gain):

• Formula: \(h = \text{Hypotenuse} \times \sin(\text{Angle})\)
• Thus, \(h = 2830 \times \sin(14.6^{\circ})\)

This calculation gives the vertical height the skier rises. It's a practical example of how trigonometry can be used to solve real-world physics problems by breaking down complex motions and paths into understandable mathematical components. Understanding these concepts reinforces students' ability to tackle diverse scenarios both in academic exercises and real-world applications.