For a physics lab experiment, four classmates run up the...

For a physics lab experiment, four classmates run up the stairs from the basement to the top floor of their physics building$-$a vertical distance of 16.0 m. The classmates and their masses are: Tatiana, 50.2 kg; Bill, 68.2 kg; Ricardo, 81.8 kg; and Melanie, 59.1 kg. The time it takes each of them is shown in $\textbf{Fig. P6.92}$. (a) Considering only the work done against gravity, which person had the largest average power output? The smallest? (b) Chang is very fit and has mass 62.3 kg. If his average power output is 1.00 hp, how many seconds does it take him to run up the stairs? Figure P6.92 (Figure can't copy)

Key Concepts

Relationship Between Energy, Time, and Power

This key concept involves understanding the direct relationship given by P = Energy / Time. In the experiment, as all subjects climb the same vertical distance (and therefore gain the same gravitational potential energy for a given mass), variations in the time taken directly affect the average power output. By rearranging this relationship, one can solve for unknown quantities, such as time needed when power and energy are known.

Gravitational Potential Energy

This concept refers to the energy an object possesses due to its position in a gravitational field. In the context of this question, it is the work done against gravity when raising a mass through a vertical height, calculated as m * g * h. This provides the basis for determining the energy expended during the stair climb.

Average Power

Average power is the rate at which work is done or energy is transferred over time. It is defined by the equation P = Work / Time. In this scenario, by comparing the work done (i.e., the increase in gravitational potential energy) and the time taken, one can determine each person's power output during the climb.

Unit Conversion

In many physics problems, it is necessary to convert between different units to maintain consistency. For example, in this question, horsepower is mentioned, so converting 1.00 hp to watts (approximately 746 watts) is essential to compute the corresponding time or power output in standard SI units.

Transcript

00:02 Once again, welcome to another problem involving power.

00:10 So there's going to be power involved.

00:13 We also have time and we also have work or energy.

00:19 We have four classmates that are running up the stairs and the height happens to be 16 .0 meters.

00:33 So they're going up the staircase, which is 16 .0 meters.

00:39 There's a graph that's given, and each one of them is shown, you know, the time in seconds, so the time it takes to go up the staircase.

00:51 The first one is tatiana and tatiana happens to be taking, you know, round about 30 seconds.

01:03 And then we have bill.

01:13 So, tatiana takes 30 seconds.

01:19 Bill takes more than tatiana.

01:26 So bill takes 50, round about 50 seconds.

01:30 This is 50.

01:33 Then ricardo, ricardo, takes about 45 seconds.

01:50 And then the last one is melanie, you know, melanie happens to take, around about 52 seconds.

01:58 So slightly above beal.

02:02 This is melanie.

02:06 Melanie takes about 52 seconds right there.

02:12 So we have four classmates running up a staircase and the staircase is has a height of 16 meters.

02:19 This is beal.

02:21 This is ricardo.

02:24 And then this is melanie.

02:31 And, you know, they all of them have different weights tautiana so the weight of tautiana empty is or not the weight but the mass of tautiana is 50 .2 kilograms the mass of beale happens to be 68 .2 kilograms the mass of ricardo happens to be 81 .8 kilograms and then the mass of melanie happens to be 59 .1 kilograms.

03:10 So we want to we want to compare the power output, you know to see who has the largest power output remember power is work over time tatiana's time you can see is 30 seconds bills time so tatiana's time is 30 seconds bill's time is 50 seconds and then ricardo's time i'm just gonna put ricardo somewhere close let me simplify this and get rid of this so ricardo's ricardo's time happens to be t ricardo tr is 45 seconds and then melanie's time is 52 seconds.

04:21 The reason why we need these times is because power is work over time.

04:29 And so if we compute the power for tatiana, we're going to take the work for tatiana.

04:38 The work tatiana does to go up the staircase over the time it takes tatiana to go up the staircase.

04:45 So that's mtgh over tt.

04:51 So we have 50 .2 kilograms times 9 .8 meters per second squared times the height which is 16 .0 meters and then divide by 30 seconds and that gives us, you know, the power output for tertiana is approximately 262 .38 watts.

05:26 Then we also have the power output for, so i'm just going to do that in the next page.

05:31 Power output for, we'll call that heat power output for for bill, would be the work done by bill and going up the staircase and the time bill takes in going up the staircase.

05:50 So that's mass of bill times gh of time for bill and that becomes the massive bill if i go back i get 68 .2 kilograms, 68 .2 kilograms times 9 .8 meters per second squared times the 16, which is the standard height, all over the time, which is 50 seconds.

06:17 And if we plug in those numbers, we get 213 .87 watts.

06:24 So we want to compare each one of them.

06:28 That's bill, the powerful bill.

06:30 This one is.

06:31 This one is.

06:31 Is the powerful tatiana.

06:35 In the next slide, we're going to do the powerful, so we had bill and tatiana.

06:40 Now we're going to do the power for ricardo.

06:42 That's the work ricardo does in going up the staircase.

06:47 As you can see, ricardo has a mass of 81 .81 kilograms.

06:53 So we want to find out...