Question

A 53 kg person eats a 518 Calorie (518 kcal) jelly doughnut for breakfast. (a) How many joules of energy are the equivalent of this jelly doughnut? (b) How many steps must this person climb on a very tall stairway to change the gravita- tional potential energy of the person-Earth system by a value equivalent to the food energy in one jelly doughnut? Assume the height of a single stair is h = 15 cm. (c) If the human body is only 22% efficient in converting chemical potential energy to mechanical energy, how many steps must the person climb to work off this delicious, amazing breakfast, and how tall of a staircase does this correspond to?

Answer 1

A 53 kg person would need to climb approximately 6357 steps to work off the energy equivalent of a 518 Calorie jelly doughnut, assuming the human body is 22% efficient in converting chemical potential energy to mechanical energy.

Step-by-step explanation:

First, let's convert the energy of the jelly doughnut from Calories (kcal) to joules (J). One Calorie is equivalent to 4184 joules, so the energy of the jelly doughnut is 518 Calories * 4184 J/Calorie = 2,165,312 joules (rounded to the nearest joule).

To determine the number of steps needed to change the gravitational potential energy of the person-Earth system by the energy equivalent of one jelly doughnut, we can use the formula:
Gravitational potential energy = mass * gravitational acceleration * height

Since each step is 15 cm in height, we can calculate the number of steps needed:
Number of steps = Gravitational potential energy / (mass * gravitational acceleration * height per step)
Number of steps = (2,165,312 J) / (53 kg * 9.8 m/s^2 * 0.15 m) = 2584 steps (rounded to the nearest step).

To calculate the number of steps needed to work off the breakfast, we need to account for the 22% efficiency of the human body in converting chemical potential energy to mechanical energy. We can use the same formula as before, but with the adjusted energy value:
Number of steps = (2,165,312 J) / (0.22 * 53 kg * 9.8 m/s^2 * 0.15 m) = 6357 steps (rounded to the nearest step).

Answer 2

A. 2,168,399.8 J

B. 27832 steps

C. 918.5 m with 6123 steps

Step-by-step explanation:

A. Since 1 cal equals to 4.1868 joules

`E=518\,kcal*(1000\,cal)/(1\,kcal) *(4.1868\,J)/(cal) =2,168,399.8\,J`

B. The change in the gravitational potential energy of the system is given by

`U_(G) =mgz`

Being m the mass of the person, g the gravity constant and z the height difference. The required z is then given by:

`z=(U_(G) )/(mg) = (2,168,399.8\,J)/(53\,kg*9.8\,m/s^(2) ) =4174.8 \,m`

Changed to 0.15 m stairs:

`steps=(4174.8\,m)/(0.15\,m) =27832\,steps`

C. At 22% efficiency, the values are reduced, since the process is less efficient:

`steps=27832\,steps*0.22=6123`

`z=4174.8\,m*0.22=918.5\,m`