Chapter 10, Problem 95A

(a)

To determine

The sprinter’s average power for the race.

Answer to Problem 95A

The sprinter’s average power for the race is $6.1\times {10}^2\text{ W}$ .

Explanation of Solution

Given:

The mass of sprinter is $m=75\text{ kg}$ .

The displacement is $d=50.0\text{ m}$

The time period is $t=8.5\text{s}$ .

Formula used:

The expression for the work done is

  $W=Fd$

Here, $F$ is the force applied and $d$ is the displacement.

The expression for the power is

  $P=\frac{W}{t}$

Here, $t$ is the time period.

The expression for the acceleration in case of zero initial velocity is

  $a=\frac{2d}{t^2}$

Calculation:

The acceleration of the sprinter as follows:

  $\begin{array}{l}a=\frac{2d}{t^2}\\ a=\frac{2\times 50.0\text{ m}}{{\left(8.5\text{ s}\right)}^2}\\ a=1.38\text{m}/{\text{s}}^2\end{array}$

The sprinter’s average power for the race as follows:

  $\begin{array}{l}P=\frac{W}{t}\\ P=\frac{Fd}{t}\\ P=\frac{mad}{t}\\ P=\frac{\left(75\text{ kg}\right)\times \left(1.38\text{m}/{\text{s}}^2\right)\times \left(50.0\text{ m}\right)}{\left(8.5\text{ s}\right)}\\ P=6.1\times {10}^2\text{ W}\end{array}$

Conclusion:

Therefore, the sprinter’s average power for the race is $6.1\times {10}^2\text{ W}$ .

(b)

To determine

The maximum power developed by the sprinter.

Answer to Problem 95A

The maximum power developed by the sprinter is $1.2\times {10}^3\text{ W}$ .

Explanation of Solution

Given:

The mass of sprinter is $m=75\text{ kg}$ .

The displacement is $d=50.0\text{ m}$

The time period is $t=8.5\text{s}$ .

Formula used:

The power is

  $\begin{array}{l}P=\frac{W}{t}\\ P=\frac{Fd}{t}\\ P=Fv\end{array}$

Calculation:

From part (a), the sprinter’s average power for the race is $P_{\text{avg}}=6.1\times {10}^2\text{ W}$ .

The power increases linearly from zero.

The maximum power developed by the sprinter as follows:

  $\begin{array}{l}{P}_{\max }=2P_{\text{avg}}\\ P_{\max }=2\times \left(6.1\times {10}^2\text{ W}\right)\\ P_{\max }=1.2\times {10}^3\text{ W}\end{array}$

Conclusion:

Thus, the maximum power developed by the sprinter is $1.2\times {10}^3\text{ W}$ .