A “POWER CLIMB”
A 50.0-kg marathon runner runs up the stairs to the top of Chicago’s 443-m-tall Willis Tower, the second tallest building in the United States (Fig. 6.28). To lift herself to the top in 15.0 minutes, what must be her average power output? Express your answer in watts, in kilowatts, what must be her average power output? Express your answer in watts, in kilowatts, and in horsepower.
IDENTIFY and SET UP:
We’ll treat the runner as a particle of mass m. Her average power output P_{av} must be enough to lift her at constant speed against gravity.
We can find P_{av} in two ways: (1) by determining how much work she must do and dividing that quantity by the elapsed time, as in Eq. (6.15) ([P_{\mathrm{av}}=\frac{\Delta W}{\Delta t}), or (2) by calculating the average upward force she must exert (in the direction of the climb) and multiplying that quantity by her upward velocity, as in Eq. (6.17) (P_{\mathrm{av}}=\frac{F_{\|} \Delta s}{\Delta t}=F_{\|} \frac{\Delta s}{\Delta t}=F_{\|} v_{\mathrm{av}}).
EXECUTE
(1) As in Example 6.8, lifting a mass m against gravity requires an amount of work equal to the weight mg multiplied by the height h it is lifted. Hence the work the runner must do is
W=m g h=(50.0 \mathrm{~kg})\left(9.80 \mathrm{~m} / \mathrm{s}^{2}\right)(443 \mathrm{~m}) =2.17 \times 10^{5} \mathrm{~J}
She does this work in a time 15.0 min = 900 s, so from Eq. (6.15) ([P_{\mathrm{av}}=\frac{\Delta W}{\Delta t}) the average power is
P_{\mathrm{av}}=\frac{2.17 \times 10^{5} \mathrm{~J}}{900 \mathrm{~s}}=241 \mathrm{~W}=0.241 \mathrm{~kW}=0.323 \mathrm{hp}(2) The force exerted is vertical and the average vertical component of velocity is (443 m)/(900 s) = 0.492 m/s, so from Eq. (6.17) the average power is
P_{\mathrm{av}}=F_{\|} v_{\mathrm{av}}=(m g) v_{\mathrm{av}}=(50.0 \mathrm{~kg})\left(9.80 \mathrm{~m} / \mathrm{s}^{2}\right)(0.492 \mathrm{~m} / \mathrm{s})=241 \mathrm{~W}
which is the same result as before.
EVALUATE: The runner’s total power output will be several times greater than 241 W. The reason is that the runner isn’t really a particle but a collection of parts that exert forces on each other and do work, such as the work done to inhale and exhale and to make her arms and legs swing. What we’ve calculated is only the part of her power output that lifts her to the top of the building.