91Ó°ÊÓ

The given velocity is $v_x=a\left(1-e^{-bt}\right)$

Acceleration is calculated as the derivative of velocity with respect to time.

$$\begin{gathered} \stackrel{→}{a_x}=\frac{d{v}_x}{dt} \\ =\frac{d}{dt}\left(a(1-e^{-bt}\right)) \\ =0-\left(a.-b.e^{-bt}\right) \\ =ab.e^{-bt} \end{gathered}$$

The acceleration is calculated above and substitute a=11.81 m/s and b=0.6887s^-1 :

$$\begin{gathered} \stackrel{→}{a_x}=11.81m/s×0.6887s^{-1}×e^{-0.6887t} \\ =8.31m/s^2×e^{-0.6887t} \end{gathered}$$

At t=0.00s, the value of acceleration is calculated as:

$$\begin{gathered} \stackrel{→}{a_{x0}}=8.13×e^{-0.6887×0} \\ =8.13m/s^2 \end{gathered}$$

At t=2.00s, the value of acceleration is calculated as:

$$\begin{gathered} \stackrel{→}{a_{x2}}=8.13×e^{-0.6887×2} \\ =2.05m/s^2 \end{gathered}$$

At t=4.00s, the value of acceleration is calculated as :

$$\begin{gathered} \stackrel{→}{a_{x4}}=8.13×e^{-0.6887×4} \\ =8.13×0.0636 \\ =0.52m/s^2 \end{gathered}$$

Distance is the anti derivative of velocity with respect to time.

$$\begin{gathered} \stackrel{→}{r}=∫v_{x}dt \\ =∫a(1-e^{-bt})dt \\ =at+\frac{a}{b}{e}^{-bt}+C \end{gathered}$$

Calculating the integrating constant C: at t=0, r=0

$$\begin{gathered} \stackrel{→}{r}=a.0+\frac{a}{b}{e}^{-b×0}+C \\ 0=\frac{a}{b}+C \\ C=-\frac{a}{b} \end{gathered}$$

Therefore, the distance travelled at t is given by:

$$\begin{gathered} \stackrel{→}{r}=at+\frac{a}{b}×e^{-bt}-\frac{a}{b} \\ =at+\frac{a}{b}(e^{-bt}-1) \end{gathered}$$

Since the equation of r is a transcendental equation, assume that the average time of the sprinter for 100 m dash is 10 s.

Substitute t=10s into the above equation for r:

$$\begin{gathered} \stackrel{→}{r}=11.81×10.00+\frac{11.81}{0.6887}(e^{-0.6887×10.00}-1) \\ =100.97m \end{gathered}$$

Repeat the process of hit and trial method and substitute at t=9.92 s:

$$\begin{gathered} \stackrel{→}{r}=11.81×9.92+\frac{11.81}{0.6887}(e^{-0.6887×9.92}-1) \\ =100.03m \end{gathered}$$

Therefore, at t=9.92 s, the length of the dash is 100m.

Therefore, the time Lewis needed to sprint 100.0 m is 9.92 s.