Problem 82 - University Physics with Modern Physics

Question

A typical doughnut contains 2.0 g of protein, 17.0 g of carbohydrates, and 7.0 g of fat. Average food energy values are 4.0 kcal/g for protein and carbohydrates and 9.0 kcal/g for fat. (a) During heavy exercise, an average person uses energy at a rate of 510 kcal/h. How long would you have to exercise to 'work off' one doughnut? (b) If the energy in the doughnut could somehow be converted into the kinetic energy of your body as a whole, how fast could you move after eating the doughnut? Take your mass to be 60 kg, and express your answer in m/s and in km/h.

Step-by-Step Solution

Answer

(a) Exercise for 16.35 minutes. (b) Speed after eating: 44.1 m/s or 158.8 km/h.

Step 1: Calculate Energy from Protein

Each gram of protein provides 4 kcal of energy. With 2.0 g of protein in the doughnut, the energy contributed by protein is:\[ \text{Energy from protein} = 2.0 \text{ g} \times 4 \text{ kcal/g} = 8.0 \text{ kcal} \]

Step 2: Calculate Energy from Carbohydrates

Each gram of carbohydrates provides 4 kcal of energy. With 17.0 g of carbohydrates in the doughnut, the energy contributed by carbohydrates is:\[ \text{Energy from carbohydrates} = 17.0 \text{ g} \times 4 \text{ kcal/g} = 68.0 \text{ kcal} \]

Step 3: Calculate Energy from Fat

Each gram of fat provides 9 kcal of energy. With 7.0 g of fat in the doughnut, the energy contributed by fat is:\[ \text{Energy from fat} = 7.0 \text{ g} \times 9 \text{ kcal/g} = 63.0 \text{ kcal} \]

Step 4: Calculate Total Energy of the Doughnut

Add the energy contribution from protein, carbohydrates, and fat to find the total energy in one doughnut:\[ \text{Total energy} = 8.0 \text{ kcal} + 68.0 \text{ kcal} + 63.0 \text{ kcal} = 139.0 \text{ kcal} \]

Step 5: Calculate Exercise Time to Burn Doughnut

Given that the energy consumption rate during exercise is 510 kcal/h, calculate the time required to burn off the doughnut's calories:\[ \text{Time} = \frac{139.0 \text{ kcal}}{510 \text{ kcal/h}} = 0.2725 \text{ h} \]Convert this to minutes:\[ 0.2725 \text{ h} \times 60 \text{ min/h} = 16.35 \text{ min} \]

Step 6: Calculate Kinetic Energy from Doughnut

The total energy from the doughnut can be converted to kinetic energy (KE) as:\[ \text{KE} = 139.0 \text{ kcal} \times 4184 \text{ J/kcal} = 581,576 \text{ J} \]

Step 7: Determine Speed from Kinetic Energy

Use the kinetic energy formula \( KE = \frac{1}{2} m v^2 \) to find speed \( v \). Rearrange to solve for \( v \):\[ v = \sqrt{\frac{2 \cdot KE}{m}} = \sqrt{\frac{2 \cdot 581,576 \text{ J}}{60 \text{ kg}}} \approx 44.1 \text{ m/s} \]

Step 8: Convert Speed to km/h

Convert the speed from m/s to km/h:\[ 44.1 \text{ m/s} \times \frac{3600 \text{ s}}{1000 \text{ m/km}} = 158.8 \text{ km/h} \]

Key Concepts

calories

Calories are a unit of energy that are used to measure the amount of energy provided by foods. When we consume foods like doughnuts, which contain protein, carbohydrates, and fat, we bring energy into our body. Here's how this works:

• Each gram of protein and carbohydrate provides approximately 4 kcal.
• Each gram of fat provides about 9 kcal.

In the example of a doughnut with 2 g of protein, 17 g of carbohydrates, and 7 g of fat, the total calorie content is calculated by multiplying each component by its respective energy value and then summing them up:

• Protein: 2 g x 4 kcal/g = 8 kcal
• Carbohydrates: 17 g x 4 kcal/g = 68 kcal
• Fat: 7 g x 9 kcal/g = 63 kcal
• Total: 8 + 68 + 63 = 139 kcal

Understanding how these units are measured can help us better manage our energy intake and expenditure, especially during physical activities.

kinetic energy

Kinetic energy is the energy that an object possesses due to its motion. In the context of our doughnut exercise, if we could convert all the energy from the doughnut directly into kinetic energy, it would give us a certain "speed-wise energy boost."

The formula for kinetic energy is:\( KE = \frac{1}{2}mv^2 \)where \( m \) is mass and \( v \) is velocity. To find out how fast you could theoretically move using all the doughnut's energy, you follow these steps:

• Convert doughnut energy from kcal to joules: \( 139 \text{kcal} \times 4184 \text{J/kcal} = 581,576 \text{J} \)
• Use the kinetic energy formula to solve for velocity \( v \):
• \( v = \sqrt{\frac{2 \times KE}{m}} = \sqrt{\frac{2 \times 581,576 \text{J}}{60 \text{kg}}} \approx 44.1 \text{m/s} \)

So theoretically, consuming the doughnut and getting all its energy as kinetic energy would propel you at about 44.1 meters per second, which converts to roughly 158.8 km/h. This is, of course, a hypothetical and incredibly fun way to think about energy conversion in the body.

exercise

Exercise is a way for our bodies to use energy, which helps in maintaining a healthy lifestyle by balancing energy intake from food and energy expenditure through physical activity. During exercise, the body utilizes calories at a certain rate, and understanding this rate helps us plan our activities.

In the example exercise, an average person burns energy at about 510 kcal/hour during intense physical activities. If you want to "burn off" the calories from a doughnut, this is how you’d calculate it:- Total calories in the doughnut: 139 kcal- Energy burn rate: 510 kcal/hourThe time needed to burn off the doughnut can be found using the formula:\[\text{Time} = \frac{\text{Calories in Doughnut}}{\text{Calorie Burn Rate}} = \frac{139}{510} \text{ hours} \approx 0.2725 \text{ hours} \]This is equal to approximately 16.35 minutes of exercise. That means if you exercise vigorously, it would take around 16 minutes to use up the energy from just one doughnut. Engaging in regular physical activities, such as running, cycling, or dancing, is an effective way to maintain energy balance and enjoy a healthier lifestyle.