Question

As an Acapulco cliff diver drops to the water from a height of 46 m, his gravitational potential energy decreases by 25,000 J. What is the diver’s weight in newtons?

Short answer

The diver's weight is approximately 543.48 newtons.

Step-by-step solution

Understand the Formula for Gravitational Potential Energy

Gravitational potential energy ( PE_g ) is given by the formula PE_g = mgh , where m is the mass of the object, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height from which the object falls. We need to find the diver's weight, which can be expressed as m imes g .

Set Up the Equation for the Problem

Since the diver's potential energy decreases by 25,000 J as he falls from 46 m, we use the equation PE_g = m imes g imes h = 25,000 J to represent this situation.

Solve for Diver's Weight

Rearrange the equation m imes g imes h = 25,000 J to solve for m imes g, which represents the diver's weight in newtons. Substituting the values, we have:\[ m imes g = \frac{25,000 \text{ J}}{46 \text{ m}} = \frac{25000}{46} \].

Calculate the Weight

Perform the division: \[ m imes g = \frac{25,000}{46} \approx 543.48 \text{ N} \].Therefore, the diver's weight is approximately 543.48 newtons.

Key concepts

Weight Calculation

Weight is fundamentally a measure of the force exerted by gravity on an object. Essentially, the weight is the gravitational pull on the mass of the object. To calculate this, we use the formula for gravitational potential energy, which is \( PE_g = mgh \). Here, \( m \) represents mass, \( g \) is the acceleration due to gravity, and \( h \) is the height from which the object is considered.In the context of the given problem, we need to determine the diver's weight as he falls freely due to gravity. The weight can be calculated by determining the product of mass and gravitational acceleration \( m \times g \), which gives us the weight in newtons.When we rearrange our potential energy equation to solve for this product, we find:

• \( m \times g = \frac{PE_g}{h} \)
• \( m \times g = \frac{25,000 \, \text{J}}{46 \, \text{m}} \)

This results in a weight of approximately 543.48 newtons for the diver.

Energy Transformation

Energy transformation is at the heart of understanding this exercise. When the diver jumps from the 46 m height, the gravitational potential energy stored in the diver decreases and transforms into kinetic energy. This transformation conforms to the conservation of energy principle, which states that energy cannot be created or destroyed, only transformed from one type to another. In free fall, the loss in gravitational potential energy equals the gain in kinetic energy. Thus, when we calculate the diver's decrease in potential energy as 25,000 J, it indicates the amount that has been converted into kinetic energy. This process is crucial as it not only helps determine the diver's speed upon reaching the water but also enables us to understand how potential energy plays an essential role in calculating weight.

Problem-Solving Steps

Breaking down problems into structured steps makes understanding and solving them much easier, just like solving for the diver's weight. Here's how you can problem-solve this scenario:

• Understand the Variables: Identify the potential energy formula \( PE_g = mgh \), where the diver’s specific details such as energy drop (25,000 J), and the fall height (46 m) are used.
• Set Up Your Equation: Apply the known values to the equation \( PE_g = m \times g \times h = 25,000 \; \text{J} \).
• Rearrange to Isolate the Unknown: We are interested in the weight, \( m \times g \), thus rearrange the equation to solve for it, resulting in \( m \times g = \frac{25,000}{46} \).
• Calculate: Perform the calculations to find \( m \times g \), which gives the weight as 543.48 newtons.

By following these structured steps, one can approach problem-solving logically, ensuring all necessary facets are covered efficiently.