Problem 52 - University Physics with Modern Physics

Question

It is 5.0 km from your home to the physics lab. As part of your physical fitness program, you could run that distance at 10 km/h (which uses up energy at the rate of 700 W), or you could walk it leisurely at 3.0 km/h (which uses energy at 290 W). Which choice would burn up more energy, and how much energy (in joules) would it burn? Why does the more intense exercise burn up less energy than the less intense exercise?

Step-by-Step Solution

Answer

Walking burns more energy: 1,743,480 J. It burns more because it takes longer, despite being less intense.

Step 1: Determine Time Taken for Each Activity

First, calculate how long it would take to run and walk the distance. For running: \[ \text{Time}_{run} = \frac{\text{Distance}}{\text{Speed}} = \frac{5.0 \text{ km}}{10 \text{ km/h}} = 0.5 \text{ hours} \]For walking:\[ \text{Time}_{walk} = \frac{5.0 \text{ km}}{3.0 \text{ km/h}} = \frac{5.0}{3.0} \text{ hours} \approx 1.67 \text{ hours} \]

Step 2: Calculate Energy Burned for Each Activity

Use the formula Energy (in joules) \( E = P \times t \), where \( P \) is power in watts and \( t \) is time in seconds.For running:\[ \text{Time}_{run} \text{ in seconds} = 0.5 \times 3600 = 1800 \text{ seconds} \]Energy burned:\[ E_{run} = 700 \text{ W} \times 1800 \text{ s} = 1,260,000 \text{ J} \]For walking:\[ \text{Time}_{walk} \text{ in seconds} = 1.67 \times 3600 \approx 6012 \text{ seconds} \]Energy burned:\[ E_{walk} = 290 \text{ W} \times 6012 \text{ s} \approx 1,743,480 \text{ J} \]

Step 3: Compare the Energy Consumption

Now, compare the energy burned for each choice:Energy when running: \( 1,260,000 \text{ J} \).Energy when walking: \( 1,743,480 \text{ J} \).Walking burns more energy than running.

Step 4: Explain Why Less Intense Exercise Burns More Energy

Walking burns more energy because although it is less intense (lower rate of energy expenditure per second), it takes longer to cover the same distance, resulting in a higher total energy consumption.

Key Concepts

Power and Energy

When tackling questions about energy consumption during exercise, understanding the concepts of power and energy is key. Power is defined as the rate at which energy is used or transferred. It is measured in watts (W), where 1 watt equals 1 joule per second. In the exercise problem, running uses energy at a rate of 700 W, while walking uses it at 290 W.

Energy refers to the total amount of work done or exertion made over a period of time and is measured in joules (J). To determine how much energy is consumed during an exercise, you multiply the power by the time in seconds. This can be expressed with the formula: \[ E = P \times t \] where \( E \) is energy, \( P \) is power, and \( t \) is time.

Even if a higher power exercise, such as running, uses energy quickly, a longer duration in a lower power exercise like walking can result in higher total energy consumption. This is because while walking has a lower power rate, the time taken is significantly longer, leading to greater energy use overall.

Kinematics

Understanding kinematics, which deals with the motion of objects, is crucial when calculating the time taken for exercises like running or walking. The primary variables in kinematics are distance, speed, and time. The relationship is described by the formula: \[ ext{Time} = rac{ ext{Distance}}{ ext{Speed}} \] In the exercise, the distance from home to the physics lab is a constant 5.0 km. The difference lies in the speed. If running, the speed is 10 km/h; if walking, it is 3.0 km/h. This affects how long each activity takes. Running the 5.0 km takes 0.5 hours, while walking it takes approximately 1.67 hours.

By computing the time required for each mode of exercise, we can then use this information to determine the energy consumed. Kinematics provides the necessary groundwork for linking movement (speed and distance) to time, which is pivotal for the scenarios in the problem.

Work-Energy Principle

The work-energy principle is fundamental in understanding why different forms of exercise may result in varying levels of energy consumption. This principle states that the work done on an object is equal to the change in its kinetic energy. In the context of the exercise, no significant change in kinetic energy occurs since the problem deals with constant speeds.

However, the essence of work (which is the product of force and displacement) can be extended to how energy is expended over time at a given power level. When exercising, your body is continuously doing work, whether you're running or walking. More importantly, the principle implies that how long you sustain an activity (time) significantly impacts total energy consumption.

Thus, low-intensity exercises over extended periods can lead to high energy consumption because they involve continuous work spread out over time, even if each moment requires less energy. This explains why walking, though less intense, burns more energy overall compared to running a shorter duration with higher intensity.