91Ó°ÊÓ

The food calorie, equal to \(4186 \mathrm{~J},\) is a measure of how much energy is released when the body metabolizes food. A certain fruit-and-cereal bar contains 140 food calories. (a) If a 65 kg hiker eats one bar, how high a mountain must he climb to "work off" the calories, assuming that all the food energy goes into increasing gravitational potential energy? (b) If, as is typical, only \(20 \%\) of the food calories go into mechanical energy, what would be the answer to part (a)? (Note: In this and all other problems, we are assuming that \(100 \%\) of the food calories that are eaten are absorbed and used by the body. This is not true. A person's "metabolic efficiency" is the percentage of calories eaten that are actually used; the body eliminates the rest. Metabolic efficiency varies considerably from person to person.

Step by step solution

01

Calculate Total Energy from Bar

First, convert calories to joules to get the total energy. One food calorie equals 4186 J. Multiply the calorie content of the fruit-and-cereal bar by 4186 to get the total energy in that bar. 140 Calories * 4186 J/Calorie = 586040 J.

02

Calculate Energy Used for Climbing

Now assume that all of that energy goes into climbing the mountain. Set the gravitational potential energy formula, \(mgh\), equal to the total energy, 586040 J. The mass, \(m\), is 65 kg, so we isolate \(h\) to calculate.586040 J = (65 kg)(9.8 m/s^2)h

03

Solve for mountain height for part a

The next step is to solve for \(h\), giving the height a hiker has to climb to work off all the calories in one bar. We get586040 J / [(65 kg)(9.8 m/s^2)] = h

04

Calculate Energy Used With Metabolic Efficiency for part b

However, as noted in the problem, typically only 20% of the calories are converted into mechanical energy. Therefore, repeat the step 2 but with only 20% of the energy, (0.20) * 586040 J. Set this equal to \(mgh\).0.20 * 586040 J = (65 kg)(9.8 m/s^2)h

05

Solve for mountain height for part b

Finally, solve for \(h\) with the adjusted energy values for metabolic efficiency to find a new height. We get (0.20 * 586040 J) / [(65 kg)(9.8 m/s^2)] = h

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Key Concepts

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

Calories to Joules Conversion

In biology, energy is often discussed in terms of calories, especially when it comes to nutrition and metabolic processes. However, in physics, the preferred unit of energy is the joule. Converting calories to joules allows us to apply physical principles like those governing mechanical energy to biological processes.

• 1 food calorie (sometimes noted as Calorie with a capital "C") is equivalent to 4186 joules.
• To convert energy in calories to joules, simply multiply the number of calories by 4186.
• This conversion is crucial when calculating the energy used in activities like hiking, as it allows us to use equations dealing with energy in a consistent unit.

In our example, a fruit-and-cereal bar containing 140 food calories provides 586,040 joules of energy (140 Cal × 4186 J/Cal = 586,040 J). This is crucial for performing calculations related to how energy is expended or stored during physical activities.

Gravitational Potential Energy

Gravitational potential energy (GPE) is the energy that an object possesses due to its position relative to the Earth. It depends on the object's mass, the height above the ground, and the gravitational acceleration.

• The formula to calculate gravitational potential energy is: \(PE_{gravity} = mgh\)
• Where \(m\) = mass (in kilograms), \(g\) = gravitational acceleration (approximately 9.8 m/s² on Earth), \(h\) = height (in meters)

In the context of our problem, we calculate the height a hiker must climb to convert all the energy from the food bar into gravitational potential energy:

• For all energy used: \(h = \frac{586,040 \text{ J}}{65\text{ kg} \times 9.8 \text{ m/s}^2}\)

This shows the relationship between caloric energy intake and the mechanical work done in climbing, linking biological consumption with physical output.

Metabolic Efficiency

Metabolic efficiency refers to the percentage of consumed energy that is converted into useful work by the body. Not all energy from food is used productively; some is lost as heat or through other biological processes that don't contribute to mechanical work.

• Typically, only about 20% of consumed energy is converted into mechanical energy, such as that used in climbing.
• Thus, if 140 food calories are consumed, only about 20% of the resultant 586,040 joules can be used for actual work.
• This equates to \(0.20 \times 586,040 \text{ J} = 117,208 \text{ J}\) being usable for climbing.

The metabolic efficiency concept is crucial for understanding how food energy translates into physical activity. For example, with a 20% efficiency, the height calculation becomes:

• For the energy that is metabolically efficient: \(h = \frac{117,208 \text{ J}}{65 \text{ kg} \times 9.8 \text{ m/s}^2}\)

Thus, metabolic efficiency is an essential consideration for assessing how diet impacts physical capability and energy expenditure.