40) Climbing the Empire State Building A new record for running the stairs of the Empire State Building was set on February 4, 2003. The 86 flights, with a total of 1576 steps, was run in 9 minutes and 33 seconds. If the height gain of each step was 0.20 m, and the mass of the runner was 70.0 kg, what was his average power output during the climb? Give your answer in both watts and horsepower.

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AmarahO50584 AmarahO50584 · Answer 1 · 2022-10-25T16:27:51+02:00

The power is given by:

[tex]P=\frac{W}{t}[/tex]

where W is the work and t is the time.

We know that the work done is related to the gravitational potential energy by:

[tex]W=mg(y_f-y_0)[/tex]

where yf is the final height of the object and y0 is the initial height.

Now, in this case we have a total of 1576 steps, each of them with a height of 0.2 meters that means that the total height gained is:

[tex](1576m)\cdot0.2=315.2\text{ m}[/tex]

This in turns means that the work done is:

[tex]W=(70\operatorname{kg})(9.8\frac{m}{s^2})(315.2m)=216227.2\text{ J}[/tex]

Now, the time it takes to achieve this is 9 minutes 33 seconds, this is the same as:

[tex]9(60)+33=573\text{ s}[/tex]

Finally we use the power formula:

[tex]P=\frac{216227.2\text{ J}}{573\text{ s}}=377.36\text{ W}[/tex]

Now we need to remember that 1 Hp is equal to 745.7 W, then we have that this is the same as:

[tex]377.36\text{ W}\cdot\frac{1\text{ Hp}}{745.7\text{ W}}=0.506[/tex]

Therefore the power to make that climb is 377.36 W or 0.506 Hp