A certain sprinter has a top speed of 11.0 m/s. If the sprinter starts from rest and accelerates at a constant rate, he is able to reach his top speed in a distance of 12.0 m. He is then able to maintain this top speed for the remainder of a 100 m race. (a) What is his time for the 100 m race? (b) In order to improve his time, the sprinter tries to decrease the distance required for him to reach his top speed. What must this distance be if he is to achieve a time of 10.0 s for the race?

The top speed of sprinter, $\mathrm{v}=11.0\mathrm{m}/\mathrm{s}$

Distance taken to reach top speed, ${\mathrm{x}}_1=12.0\mathrm{m}$

Total distance,$\mathrm{x}=100\mathrm{m}$

Kinematic equations describe the motion of an object with constant acceleration. These equations can be used to determine the acceleration, velocity or distance.

The expression for the kinematic equations of motion are given as follows:

$\mathrm{v}={\mathrm{v}}_0+\mathrm{at}$ … (i)

$\mathrm{x}={\mathrm{v}}_0\mathrm{t}+\frac{1}{2}{\mathrm{at}}^2$ … (ii)

${\mathrm{v}}^2={{\mathrm{v}}_0}^{2}2\mathrm{ax}$ … (iii)

Here, $v_0$is the initial velocity, $v$is the final velocity, $t$is the time, $a$is the acceleration and$x$is the displacement.

Since the sprinter starts from rest therefore, its initial velocity is zero.

Using equation (iii), the acceleration can be calculated as follows:

$$\begin{gathered} \mathrm{a}=\frac{{\mathrm{v}}^2-{\mathrm{v}}^2}{2\mathrm{x}} \\ =\frac{{(11.0\mathrm{m}/\mathrm{s})}^2}{2\times 12.0\mathrm{m}} \\ =5.04\mathrm{m}/{\mathrm{s}}^2 \end{gathered}$$

Using equation (i), the time taken to achieve top speed is calculated as follows:

$$\begin{gathered} {\mathrm{t}}_1=\frac{\mathrm{v}-{\mathrm{v}}_0}{\mathrm{a}} \\ =\frac{11.0\mathrm{m}/\mathrm{s}-0\mathrm{m}/\mathrm{s}}{5.04\mathrm{m}/\mathrm{s}} \\ \approx 2.2\mathrm{s} \end{gathered}$$

Now, the time taken by sprinter for the remaining race can be calculated as follows:

$$\begin{gathered} {\mathrm{t}}_2=\frac{\left(100\mathrm{m}-12\mathrm{m}\right)}{11.0\mathrm{m}/\mathrm{s}} \\ =\frac{88\mathrm{m}}{11.0\mathrm{m}/\mathrm{s}} \\ =8.0\mathrm{s} \end{gathered}$$localid="1657713441914" $$\begin{gathered} {\mathrm{t}}_2=\frac{(100\mathrm{m}-12\mathrm{m})}{11.0\mathrm{m}/\mathrm{s}} \\ =\frac{88\mathrm{m}}{11.0\mathrm{m}/\mathrm{s}} \\ =8.0\mathrm{s} \end{gathered}$$

Now, the total time taken by the sprinter for the 100 m race is,

localid="1657713446237" $$\begin{gathered} \mathrm{t}={\mathrm{t}}_1+{\mathrm{t}}_2 \\ =8.0\mathrm{s}+2.2\mathrm{s} \\ =10.2\mathrm{s} \end{gathered}$$

Thus, total time taken by the sprinter for the 100 m race is 10.02 s .

If he takes a time of 10.0 s for the 100 m race, then he has to achieve his top speed in 10 s-8 s=2 s

Using equation (ii), the distance required to achieve top speed is calculated as follows:

$$\begin{gathered} \mathrm{x}={\mathrm{v}}_0\mathrm{t}+\frac{1}{2}{\mathrm{at}}^2 \\ =0+\frac{1}{2}(5.04\mathrm{m}/{\mathrm{s}}^2){(2\mathrm{s})}^2 \\ \approx 10.0\mathrm{m} \end{gathered}$$

Therefore, he has to achieve his top speed in 10 m .