A 91 kg student eats a 200 Calorie doughnut. To “burn it off”, he decides to climb the steps of a tall building. (1 food Calorie = 10^3 Calories.) How high would he have to climb to expend an equivalent amount of work? The acceleration of gravity is 9.8 m/s2
To determine the height the student would need to climb in order to expend an equivalent amount of work to the 200 Calorie doughnut, we can use the concept of work and potential energy.
We'll start by converting the 200 Calories into Joules. Since 1 Calorie is equal to 10^3 Calories, 200 Calories is equal to 200 x 10^3 Calories.
To convert Calories to Joules, we can use the conversion factor 1 calorie = 4.18 Joules. Therefore, 1 Calorie is equal to 4.18 x 10^3 Joules.
So, 200 Calories is equal to 200 x 10^3 x 4.18 Joules.
Next, we need to find the potential energy (PE) gained by climbing the stairs, which is equal to mgh, where m is the mass of the student (91 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height.
Equating the potential energy to the work done (200 x 10^3 x 4.18 Joules), we have:
mgh = 200 x 10^3 x 4.18 Joules
Substituting the values, we get:
91 kg * 9.8 m/s^2 * h = 200 x 10^3 x 4.18 Joules
Simplifying the equation:
891.8 h = 836,000 Joules
Now, we can solve for h by dividing both sides of the equation by 891.8:
h = 836,000 Joules / 891.8
Calculating the value:
h ≈ 936.7 meters
Therefore, the student would need to climb approximately 936.7 meters to expend an equivalent amount of work to the 200 Calorie doughnut.
To find out how high the student would have to climb to expend an equivalent amount of work, we need to calculate the amount of work done by eating the doughnut and then equate it to the work done while climbing the steps.
1. Start by converting the energy content of the doughnut from Calories to Joules, since we'll be using SI units. We know that 1 food Calorie is equal to 10^3 Calories, so:
Energy in the doughnut = 200 Calories * (10^3 Calories/1 food Calorie) = 200 * 10^3 Calories
2. Next, convert the energy in the doughnut to Joules. We know that 1 calorie is equal to 4.18 Joules, so:
Energy in the doughnut = 200 * 10^3 Calories * (4.18 J/1 calorie) = 83600 * 10^3 Joules
3. Now, we need to calculate the work done while climbing the steps. The work done is equal to the force applied multiplied by the distance traveled. In this case, the force we need to consider is the force against gravity, and the distance is the height climbed.
Work done = Force * Distance
4. To calculate the force, we'll use the weight of the student. The weight is equal to the mass multiplied by the acceleration due to gravity.
Weight of the student = mass * acceleration due to gravity
= 91 kg * 9.8 m/s^2
5. Substitute the force into the work equation:
Work done = (91 kg * 9.8 m/s^2) * Distance
6. Finally, we can equate the energy in the doughnut to the work done while climbing the steps:
83600 * 10^3 Joules = (91 kg * 9.8 m/s^2) * Distance
Now, solve for the height (distance) by rearranging the equation:
Distance = (83600 * 10^3 Joules) / (91 kg * 9.8 m/s^2)
Calculate the result to find out how high the student would have to climb.
91 * 9.8 * h = 200,000
h = 224 meters
assuming that a floor is about 3 meters, that is about a 75 story building.