First Principles AI

Solution

Given:

d(s)=0.02s^2+0.1s

Safe braking means:

0.02s^2+0.1s\le 50

Multiply by 100:

2s^2+10s\le 5000

Divide by 2:

s^2+5s-2500\le 0

Solve $s^2+5s-2500=0$ :

\begin{aligned} s&=\frac{-5\pm\sqrt{25+10000}}{2}\\ &=\frac{-5\pm\sqrt{10025}}{2}\\ &=\frac{-5\pm 5\sqrt{401}}{2} \end{aligned}

Only the positive root is physically meaningful:

s=\frac{5(\sqrt{401}-1)}{2}\approx 47.56

Since the parabola opens upward, the inequality $\le 0$ holds between the roots. With $s\ge 0$ :

\boxed{0\le s\le \frac{5(\sqrt{401}-1)}{2}}\ \text{km/h}

\boxed{0\le s\lesssim 47.56\ \text{km/h}}