Question

The food calorie, equal to 4186 J

, is a measure of how much energy is released when food is metabolized by the body. A certain brand of fruit-and-cereal bar contains 130 food calories per bar.
If a 75.0 kg
hiker eats one of these bars, how high a mountain must he climb to "work off" the calories, assuming that all the food energy goes only into increasing gravitational potential energy?
Express your answer in meters.
and
If, as is typical, only 24.0 %
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 actually not true. A person's "metabolic efficiency" is the percentage of calories eaten that are actually used; the rest are eliminated by the body. Metabolic efficiency varies considerably from person to person.)
Express your answer in meters.

Answer 1

To calculate the height a hiker must climb to "work off" the calories, we need to convert the food calories to joules and then use the gravitational potential energy formula.

Given:

Food calorie = 4186 J

Calories per bar = 130 food calories

Hiker's weight = 75.0 kg

First, let's calculate the energy content of one bar in joules:

Energy content of one bar = 130 food calories × 4186 J/food calorie

Next, we'll calculate the potential energy gained by the hiker when climbing the mountain. Since 100% of the food energy goes into increasing gravitational potential energy, we can equate the energy content of the bar to the potential energy gained:

Potential energy gained = Energy content of one bar = 130 food calories × 4186 J/food calorie

Now, we can calculate the height of the mountain using the formula for gravitational potential energy:

Potential energy gained = mgh

Where:

m = mass (75.0 kg)

g = acceleration due to gravity (approximately 9.8 m/s²)

h = height of the mountain (unknown)

Rearranging the formula, we get:

h = Potential energy gained / (mg)

Substituting the values:

h = (130 food calories × 4186 J/food calorie) / (75.0 kg × 9.8 m/s²)

Calculating the height:

h ≈ (130 × 4186) / (75.0 × 9.8) meters

Simplifying further:

h ≈ 11271880 / 7350 meters

Therefore, the hiker must climb approximately 1534 meters to "work off" the calories, assuming 100% of the food energy goes into increasing gravitational potential energy.

Now let's consider the case where only 24.0% of the food calories go into mechanical energy. In this case, we need to adjust the calculation:

Potential energy gained = (24.0% of the energy content of one bar)

h = (24.0% × 130 food calories × 4186 J/food calorie) / (75.0 kg × 9.8 m/s²)

Calculating the height:

h ≈ (0.24 × 130 × 4186) / (75.0 × 9.8) meters

Simplifying further:

h ≈ 149815.28 / 7350 meters

Therefore, when considering that only 24.0% of the food calories go into mechanical energy, the hiker would need to climb approximately 20.38 meters to "work off" the calories.