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Physics problem solving is all about understanding the concepts involved, and then applying them step by step to reach a solution. It begins by clearly identifying what the problem is asking. In our given example, we need to determine how the gravitational potential energy of an 18.0 kg child increases when climbing to the second floor. We already know that the gravitational potential energy of an 81.0 kg adult increases by 2000 J when climbing the same height. Therefore, the first step is to extract all relevant data from the problem statement.

Next, we need to organize this data and then form a plan, often by writing down relevant formulas or principles connected to the problem, such as the formula for gravitational potential energy, which is given by:

• The gravitational potential energy formula is: \( PE = mgh \), where \( m \) is the mass, \( g \) is the acceleration due to gravity, and \( h \) is the height.

By isolating each part of the problem, and using known physics equations, we can systematically solve the problem, one piece at a time.

Energy conservation is a principle in physics that states energy cannot be created or destroyed, but can only be transformed from one form to another. This principle helps us understand that when the adult climbs the stairs to increase his gravitational potential energy by 2000 J, this energy must come from somewhere—most likely, the work done by the muscles during the climb.

In our problem, the adult uses his energy to change his position, increasing his potential energy. Similarly, the child going up the staircase also converts energy into gravitational potential energy. Even though the mass is different, the concept of energy conservation helps us determine how potential energy changes with different masses. The key takeaway is:

• Energy changes form but the total energy remains consistent.
• By maintaining the principle of energy conservation, we can predict outcomes in different scenarios, such as different masses climbing the same height.

Potential energy calculation often boils down to applying the formula correctly. To solve the problem of finding how much the potential energy of a child increases, we follow the given problem’s steps to find the height first, then use it for our calculations.

The calculated height from the adult's data gives us the basis:

• The adult's potential energy formula was rearranged to find the height: \[ h = \frac{2000}{81 \times 9.81} \approx 25.21 \, m \]
• This height is then used with the child's mass in the formula \( PE = mgh \).

To find the child's potential energy, substitute the child's mass \( m = 18.0 \, kg \) into the formula with the same height:

• Calculate: \[ PE = 18.0 \, kg \times 9.81 \, m/s^2 \times 25.21 \, m \approx 4453.638 \, J \]

This calculation reflects the increase in potential energy as a result of the child climbing the stairs, emphasizing the importance of understanding every variable involved in the formula.