Answer
The distance required to reach top speed is $~~10~m$
Work Step by Step
Let the distance required to reach top speed be $x$.
The average speed over this distance $x$ is $6.5~m/s$
We can find the time required to complete this distance $x$:
$t_1 = \frac{x}{6.5} = \frac{2x}{11}$
We can find the time required to run the rest of the race:
$t_2 = \frac{100-x}{11}$
The total time must be $10.0~s$
We can find $x$:
$t_1+t_2 = 10.0$
$\frac{2x}{11}+\frac{100-x}{11} = 10.0$
$\frac{2x+100-x}{11} = 10.0$
$2x+100-x = 110$
$x+100 = 110$
$x = 10~m$
The distance required to reach top speed is $~~10~m$.
