Question

(a) What is the power output in watts and horsepower of a 70.0-kg sprinter who accelerates from rest to 10.0 m/s in 3.00 s?

(b) Considering the amount of power generated, do you think a well-trained athlete could do this repetitively for long periods of time?

Short answer

(a) The power output of the sprinter is\(1166.67{\rm{ W}}\)or\(1.56{\rm{ hp}}\).

(b) Probably not, since a human is much less powerful.

Step-by-step solution

Work-energy theorem

According to work-energy theorem, when some work done on a moving body will result in change in the kinetic energy of the body. In other words, the change in kinetic energy of the body equals to the amount of the work done on it.

Power output in watts and horsepower

(a)

According to work-energy theorem, the work done by the sprinter is,

\(\begin{aligned}W &= \Delta {\rm{KE}}\\ &= \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2\end{aligned}\)

Here, m is the mass of the sprinter\(\left( {m = 70.0{\rm{ kg}}} \right)\),\({v_f}\)is the final velocity of the sprinter\(\left( {{v_f} = 10{\rm{ m}}/{\rm{s}}} \right)\), and\({v_i}\)is the initial velocity of the sprinter (\({v_i} = 0\)as the sprinter starts from rest).

Putting all known values,

\(\begin{aligned}W &= \frac{1}{2} \times \left( {70.0{\rm{ kg}}} \right) \times {\left( {10.0{\rm{ m}}/{\rm{s}}} \right)^2} - \frac{1}{2} \times \left( {70.0{\rm{ kg}}} \right) \times {\left( 0 \right)^2}\\ &= 3500{\rm{ J}}\end{aligned}\)

The power output of the sprinter is,

\(P = \frac{W}{t}\)

Here, t is the time required to attain the final velocity\(\left( {t = 3.00{\rm{ s}}} \right)\).

Putting all known values,

\(\begin{aligned}P& = \frac{{3500{\rm{ W}}}}{{3.00{\rm{ s}}}}\\ &= 1166.67{\rm{ W}}\end{aligned}\)

Converting power output of the sprinter from watts to horsepower,

\(\begin{aligned}P& = \left( {1166.67{\rm{ W}}} \right) \times \left( {\frac{{1{\rm{ hp}}}}{{746{\rm{ W}}}}} \right)\\ &= 1.56{\rm{ hp}}\end{aligned}\)

Therefore, the required power output of the sprinter is \(1166.67{\rm{ W}}\) or \(1.56{\rm{ hp}}\).

Comparing power output

(b)

No, a well-trained athlete cannot give this much power output for a longer time because a human is much less powerful. He cannot perform with so much power for longer period of time.