Question

Yosemite Falls in California is about 740 m high. (a) What average power would it take for a 70 -kg person to hike up to the top of Yosemite Falls in $1.5 \mathrm{h} ?\( (b) The human body is about \)25 \%$ efficient at converting chemical energy to mechanical energy. How much chemical energy is used in this hike? (c) One food Calorie is equal to \(4.186 \times 10^{3} \mathrm{J} .\) How many Calories of food energy would a person use in this hike?

Short answer

Question: Determine the average power, chemical energy, and Calories of food energy used by a 70-kg person hiking up Yosemite Falls (740m high) in 1.5 hours, assuming a 25% efficiency of the human body. Answer: The hiker uses an average power of 940.27 W, chemical energy of 20,309,840 J, and 4853.95 Calories of food energy during the hike.

Step-by-step solution

Calculate the gravitational potential energy

To find the average power required to hike up Yosemite Falls, we first need to calculate the gravitational potential energy gain of the person. This is given by the formula \(PE = mgh\), where \(PE\) is the gravitational potential energy, \(m\) is the mass of the person, \(g\) is the acceleration due to gravity \((9.81 m/s^2)\), and \(h\) is the height climbed. So, \(PE = 70\,\text{kg} \times 9.81\,\frac{\text{m}}{\text{s}^2} \times 740\,\text{m} = 5077460\,\text{J}\).

Calculate the average power needed for the hike

Now, we need to find the average power required for the hike. Taking into account that 1.5 hours is equal to 1.5 x 3600 = 5400 seconds. Power is the rate of energy use, so \(P = \frac{PE}{t}\), where \(P\) is the average power and \(t\) is the time taken. Therefore, \(P = \frac{5077460\,\text{J}}{5400\,\text{s}} = 940.27\,\text{W}\).

Calculate the chemical energy used during the hike

The human body is not 100% efficient in converting the food energy to mechanical energy. Since the body is only 25% efficient in converting food energy, we can find the amount of chemical energy used: \(Chemical\,Energy = \frac{5077460\,\text{J}}{0.25} = 20309840\,\text{J}\).

Calculate the food energy used during the hike

Finally, we need to find the number of Calories consumed during the hike. We know that 1 food Calorie is equal to 4.186 x 10^3 J. So, we can find the number of Calories used by converting the chemical energy to Calories: \(Food\,Energy = \frac{20309840\,\text{J}}{4.186\times10^3\,\frac{\text{J}}{\text{Cal}}} = 4853.95\,\text{Cal}\). To summarize: (a) The average power required for a 70-kg person to hike up Yosemite Falls in 1.5 hours is 940.27 W. (b) The amount of chemical energy used during the hike is 20,309,840 J. (c) The number of Calories of food energy used during the hike is 4853.95 Calories.