Question

Marines of bodyweight \(70[\mathrm{~kg}]\) reach an altitude of \(2923[\mathrm{~m}]\) in \(7.75\) hours. Calculate the total potential energy gained. What is the resulting power? What is more sensible, to calculate the power based on \(7.75\) hours, or based on 24 hours?

Short answer

Total potential energy gained is 2,004,074 J. Power based on 7.75 hours is 71.84 W, based on 24 hours is 23.2 W. It's more sensible to use 7.75 hours for power calculation.

Step-by-step solution

Understanding the Problem

We need to calculate both the total potential energy gained and the resulting power of marines reaching an altitude of 2923 m with a bodyweight of 70 kg in 7.75 hours. Then, decide how to sensibly calculate the power.

Calculate Potential Energy

The potential energy gained by an object is given by the formula \( PE = mgh \), where \( m = 70 \, \text{kg} \) is the mass, \( g = 9.8 \, \text{m/s}^2 \) is the acceleration due to gravity, and \( h = 2923 \, \text{m} \) is the height. Substituting these values we get:\[ PE = 70 \times 9.8 \times 2923 = 2004074 \, \text{J} \].

Calculate the Power Based on 7.75 Hours

Power is the rate of work done or energy transferred over time, given by \( P = \frac{E}{t} \). First, convert 7.75 hours to seconds: \( 7.75 \times 3600 = 27900 \, \text{s} \). Then, calculate the power: \[ P = \frac{2004074}{27900} = 71.84 \, \text{W} \].

Calculate the Power Based on 24 Hours

To calculate power based on 24 hours, convert the time to seconds: \( 24 \times 3600 = 86400 \, \text{s} \). Then, calculate the power using the same energy value:\[ P = \frac{2004074}{86400} = 23.2 \, \text{W} \].

Evaluating the Sensible Approach

To determine what is more sensible, consider the context. Since the marines took 7.75 hours to reach the altitude, using this time for power calculation better represents the physical effort and actual rate of energy expenditure. Using 24 hours understates the intensity of the task over the duration it was performed.

Key concepts

Power Calculation

Power is a measure of how quickly energy is used or transformed over a given time. In the context of this exercise, marines climbing to an altitude means they are using their body's energy stores to gain potential energy. The formula to calculate power ( P) is:

• \( P = \frac{E}{t} \), where \( E \) is energy in joules, and \( t \) is time in seconds.

To find the power based on the 7.75 hours it took the marines to ascend:

• Convert the time from hours to seconds: \( 7.75 \times 3600 = 27900 \, \text{s} \).
• Use the potential energy calculated: 2004074 J.
• Substitute these values into the formula: \( P = \frac{2004074}{27900} = 71.84 \, \text{W} \).
• "W" or watts is the unit of power, which tells us how much work is being done per second.

It is common to convert time into seconds for precision since power reflects momentum per second. This calculation shows that 71.84 watts of energy were exerted by the marines while climbing in 7.75 hours.

Rate of Energy Expenditure

The rate of energy expenditure is about how fast energy is used over a time period. When we talk about this in terms of physical activities like climbing, it describes how quickly someone uses energy to perform the task. When the exercise asks to calculate power for different time periods (7.75 hours and 24 hours), it is essentially asking how the rate of energy expenditure changes with time considerations.For the 24-hour scenario:

• The formula remains: \( P = \frac{E}{t} \).
• Convert 24 hours to seconds: \( 24 \times 3600 = 86400 \, \text{s} \).
• Calculate power: \( P = \frac{2004074}{86400} = 23.2 \, \text{W} \).

Comparing the power calculations of 23.2 W for 24 hours and 71.84 W for 7.75 hours illustrates how a longer time frame smooths out the energy expenditure, distributing it more evenly. However, focusing on the actual activity duration (7.75 hours) provides a clearer picture of how intense the energy use was, showcasing the real effort.

Physical Effort

Physical effort can be understood as the amount of energy exerted during activities like climbing. It is closely tied to concepts like potential energy and power, both of which describe the transformation and expenditure of energy. When the marines climbed to a high altitude, they converted stored energy into work, requiring effort. Potential energy gained reflects the work done against gravity. By focusing on the time it took to perform the ascent (7.75 hours), we determine how rigorous and intense the physical effort was. It's important to recognize that:

• Using the entire 24-hour day might miss unique exertions or intense moments during that time.
• The amount of work done in shorter durations can better reflect the physical demand placed on the body.

Thus, for activities demanding intense or vigorous effort, assessing power over the actual duration (7.75 hours) gives a more accurate depiction of physical effort endured compared to spreading it over longer periods, which may dilute the effort's intensity.