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Chapter 6: Problem 64

Lars, of mass \(82.4 \mathrm{kg},\) has been working out and can do work for about 2.0 min at the rate of \(1.0 \mathrm{hp}(746 \mathrm{W})\) How long will it take him to climb three flights of stairs, a vertical height of $12.0 \mathrm{m} ?$

Short Answer

Expert verified
Answer: It takes the student 13 seconds to climb the three flights of stairs, which is within his 2.0-minute ability to work at a rate of 1.0 hp.

Step by step solution

01

Calculate gravitational potential energy gained by the student while climbing the stairs

We can calculate gravitational potential energy using the formula: \(E_p = mgh\), where \(E_p\) is the gravitational potential energy, \(m\) is the mass (82.4 kg), \(g\) is the acceleration due to gravity (9.81 \(m/s^{2}\)), and \(h\) is the height (12.0 m). \(E_p = (82.4 \mathrm{kg})(9.81 \mathrm{m/s^2})(12.0 \mathrm{m}) = 9704.448 \mathrm{J}\) (Joules)
02

Calculate total work the student can do in 2.0 minutes

Since the student can work at a rate of 1.0 hp (746 W) for 2.0 minutes, we first need to convert 2.0 minutes into seconds. There are 60 seconds in a minute, so: Time = \(2.0 \times 60 = 120 \ \mathrm{s}\) (seconds) Now calculate the total work the student can do in 2.0 minutes: Work = Power x Time Work = \(746 \ \mathrm{W} \times 120\ \mathrm{s} = 89520 \mathrm{J}\)
03

Calculate the time it takes the student to climb the stairs

We will use the power formula to determine the time it takes for the student to climb the stairs: Power = Work / Time Rearranging the formula to solve for time: Time = Work / Power Since we found in step 1 that the work needed to climb the stairs (gravitational potential energy gained) is 9704.448 J, and we know the power at which the student works (1.0 hp or 746 W): Time = \(\dfrac{9704.448 \ \mathrm{J}}{746 \ \mathrm{W}} = 13 \ \mathrm{s}\) It will take the student 13 seconds to climb the three flights of stairs, which is within his 2.0-minute ability to work at a rate of 1.0 hp.

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