Question

A \(65-\mathrm{kg}\) hiker climbs to the second base camp on Nanga Parbat in Pakistan, at an altitude of \(3900 \mathrm{~m}\), starting from the first base camp at \(2200 \mathrm{~m}\). The climb is made in \(5.0 \mathrm{~h}\). Calculate (a) the work done against gravity, (b) the average power output, and (c) the rate of energy input required, assuming the energy conversion efficiency of the human body is \(15 \%\)

Short answer

input_energy = ΔPE / 0.15 Now, we substitute the values into the formula to find the rate of energy input required: input_energy = 1,095,645 J / 0.15 = 7,304,300 J To find the rate of energy input, we divide the input_energy by the time in seconds: Rate of energy input = 7,304,300 J / 18,000 s = 405.8 W The rate of energy input required is 405.8 Watts.

Step-by-step solution

Calculate the change in gravitational potential energy

The gravitational potential energy (PE) of an object is given by the formula PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height above a reference point. In this case, the change in gravitational potential energy will be equal to the work done against gravity. First, we need to calculate the change in altitude (Δh) by finding the difference between the final altitude (3900 m) and the initial altitude (2200 m): Δh = 3900 m - 2200 m = 1700 m Now we can calculate the change in gravitational potential energy using the given mass (m = 65 kg) and the acceleration due to gravity (g ≈ 9.81 m/s^2): ΔPE = m * g * Δh

Calculate the work done against gravity

Now we substitute the values into the formula to find the work done against gravity: ΔPE = 65 kg * 9.81 m/s² * 1700 m = 1095645 J The work done against gravity is 1,095,645 Joules.

Calculate the average power output

Power is defined as the rate at which work is done or energy is transferred. We can calculate the average power output by dividing the work done against gravity by the time taken to climb: P_avg = ΔPE / time The given time is 5.0 hours, but we need the time in seconds to match the work done in Joules. Time (in seconds) = 5.0 h * 3600 s/h = 18000 s Now, we can calculate the average power output: P_avg = 1,095,645 J / 18,000 s = 60.87 W The average power output is 60.87 Watts.

Calculate the rate of energy input required

Finally, we need to determine the rate of energy input required, taking into account the energy conversion efficiency of the human body. The given efficiency is 15%, so the hiker's body converts only 15% of the input energy into work. Therefore, we can set up a proportion: (input_energy * 0.15) = ΔPE \\

Key concepts

Work Done Against Gravity

Gravity is a constant force that objects with mass exert on each other, and it becomes particularly noticeable when dealing with motions on Earth. When you elevate an object against Earth's gravitational pull, it requires energy. This is the 'work done against gravity.' In our hiker's scenario, climbing from a lower altitude to a higher one is literally an uphill battle against gravity.

The hiker's work done is quantified by the change in gravitational potential energy, which is the energy an object possesses due to its position in a gravitational field. The formula for gravitational potential energy is \( PE = mgh \), where \( m \) is the mass in kilograms, \( g \) is the acceleration due to gravity (approximated as \( 9.81 m/s^2 \) on Earth), and \( h \) is the height in meters. Since 'work' and 'gravitational potential energy' in this context are interchangeable, the work done by the hiker against gravity is essentially the energy needed to go from the first base camp to the second base camp on Nanga Parbat. This is calculated using the difference in altitude, and for our hiker, the work done against gravity is substantial, amounting to 1,095,645 Joules.

Average Power Output

Power relates to how quickly work is done or energy is transferred. When calculating the 'average power output' in any physical activity, such as a hike, we're essentially determining the hiker’s sustained capacity to do work over a period.

The average power output is given by the formula \( P_{avg} = \Delta PE / time \), where \( \Delta PE \) is the work done against gravity, and \( time \) is the duration over which the work is done. In our case, after converting 5 hours into seconds (to match the units of the energy exerted in Joules), we find that the hiker has an average power output of 60.87 Watts. This measure gives us an idea of the rate at which the hiker has done work against the gravitational force while climbing the mountain.

• Maintaining this steady power output over several hours highlights the endurance and physical condition required for such a task.

Energy Conversion Efficiency

When human beings perform physical activities, not all the energy consumed is used for work. This is where 'energy conversion efficiency' comes into play. It's a measure of how effectively energy input, such as food in this case, is converted into work output. In real-world scenarios, a large amount of energy is lost as heat or used for other bodily functions.

The hiker's energy conversion efficiency is only 15%, quite low, which means that 85% of the hiker's energy intake is not going towards doing the work of climbing but is dissipated in other ways. To determine the total energy that must be consumed for the entire hike, you would need to take the work done against gravity and account for this efficiency. The relationship is given as \( \text{input energy} \times \text{efficiency} = \Delta PE \).

• If we were to calculate the actual energy intake required for the hiker's climb, considering the energy conversion efficiency, it would be much higher than the work done against gravity alone, reflecting the body's multifaceted energy demands during strenuous tasks.