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On flat ground, a \(70 \mathrm{~kg}\) person requires about \(300 \mathrm{~W}\) of metabolic power to walk at a steady pace of \(5.0 \mathrm{~km} / \mathrm{h}(1.4 \mathrm{~m} / \mathrm{s})\) Using the same metabolic power output, that person can bicycle over the same ground at \(15 \mathrm{~km} / \mathrm{h}\). A \(70 \mathrm{~kg}\) person walks at a steady pace of \(5.0 \mathrm{~km} / \mathrm{h}\) on a treadmill at a \(5.0 \%\) grade (that is, the vertical distance covered is \(5.0 \%\) of the horizontal distance covered). If we assume the metabolic power required is equal to that required for walking on a flat surface plus the rate of doing work for the vertical climb, how much power is required? A. \(300 \mathrm{~W}\) B. 315 W C. \(350 \mathrm{~W}\) D. \(370 \mathrm{~W}\)

Step by step solution

01

Identify the given values

To start, identify the given values in the problem:- Person's mass, \( m = 70 \text{ kg} \)- Metabolic power for walking on level ground, \( P_0 = 300 \text{ W} \)- Speed of walking, \( v = 5.0 \text{ km/h} = 1.4 \text{ m/s} \)- Treadmill grade, \( 5.0\% \) or \( 0.05 \) slope.

02

Determine the component of speed contributing to vertical work

The treadmill grade tells us the slope is 5%, meaning the vertical component of speed is based on this slope. We calculate it as follows:\[ v_y = v \times \text{slope} = 1.4 \text{ m/s} \times 0.05 = 0.07 \text{ m/s} \]

03

Calculate gravitational power required to maintain this climb

Use the formula for power to maintain the vertical movement: \( P_{climb} = m \cdot g \cdot v_y \), where \( g = 9.8 \text{ m/s}^2 \) (acceleration due to gravity).\[ P_{climb} = 70 \text{ kg} \times 9.8 \text{ m/s}^2 \times 0.07 \text{ m/s} = 48.02 \text{ W} \]

04

Add metabolic power and climbing power

To find the total metabolic power required, add the power for flat walking and the additional power for climbing:\[ P_{total} = P_0 + P_{climb} = 300 \text{ W} + 48.02 \text{ W} = 348.02 \text{ W} \]

05

Choose the closest answer option

Now, choose the closest answer option from the given choices. The calculated power is approximately 348 W, which is closest to 350 W in the options provided.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. When discussing a person walking or running, it's crucial to understand how their movement relates to kinetic energy. The basic formula for kinetic energy (KE ) is given by:\[ KE = \frac{1}{2} m v^2 \]where

• \( m \) is the mass of the person (in kilograms), and
• \( v \) is the velocity (in meters per second).

In the context of exercises involving walking or cycling, such as the one discussed, we consider how their mass and speed contribute to the overall energy used to maintain motion. It's important to understand that the kinetic energy component is a part of the total energy expenditure, which affects the metabolic power required. Remember, when a person walks faster, their kinetic energy increases as velocity is squared in the kinetic energy equation.

Vertical Climb

Vertical climb refers to the upward movement against gravity. When a person is walking on an incline, such as a 5treadmill with a 5% grade, they are constantly climbing vertically relative to the horizontal distance covered. The vertical climb requires additional metabolic power compared to walking on a flat surface.
To calculate the vertical component of speed, multiply the total speed by the slope percentage. In our exercise, this was calculated as:\[ v_y = 1.4 \text{ m/s} \times 0.05 = 0.07 \text{ m/s} \]This vertical speed is essential for determining the climbing power required. It's a straightforward yet important part of solving problems involving inclined walking, as it directly impacts the additional energy expenditure needed to overcome the elevation difference.

Gravitational Work

Gravitational work is the work done against the force of gravity. When moving upwards, such as walking on an incline, more energy is spent against gravity, which requires additional power. The gravitational power needed is calculated using the formula:\[ P_{climb} = m \cdot g \cdot v_y \]where

• \( m \) is the mass of the person,
• \( g \) is the acceleration due to gravity (approximately 9.8 m/s^2), and
• \( v_y \) is the vertical component of velocity.

In the given scenario, the calculated gravitational power to maintain the vertical climb was 48.02 W. This value is then added to the base metabolic power required for level ground walking to find the total power necessity. Gravitational work becomes a primary factor when considering activities involving elevation, such as hiking or treadmill exercises.

Treadmill Dynamics

Treadmill dynamics involve understanding how the machine simulates outdoor conditions like flat or incline walking. A treadmill set at a 5% grade forces the person to work harder than on flat ground by requiring additional energy to move upwards. The treadmill dynamics affect the metabolic power needed. In this exercise:

• The treadmill requires the same basal metabolic power as flat walking because the speed remains constant.
• Additional metabolic power is required for the vertical component.

To calculate the total percent of power consumed, you sum the metabolic power for walking on level terrain with the power required for the vertical climb, arriving at approximately 348 W in this case. Hence, treadmill dynamics are intricate as they blend standard walking mechanics with additional exertion due to incline, significantly impacting the overall energy expenditure.