# A car can accelerate to a Maximum speed $M$ at $1.5 m/s^2$ and brake at $2.0 m/s^2$…

So, the problem goes thusly: a car can accelerate to a maximum speed of $M$, at $1.5 \space \mathrm{m/s}^2$ and brake at $2 \space \mathrm{m/s}^2$. How long does it take to go a thousand kilometers?

Also, for my own personal edification, I asked: what speed can the car get up to such that it can still stop within a thousand meters?

My approach for the acceleration goes something like this:

$$a=1.5 \space \mathrm{(m/s)}^2 \\v=1.5t \\M=1.5t \space \mathrm{so} \space t=\frac{M}{1.5} \\s_0=.75t^2=.75(\frac{M}{1.5})^2$$ $\frac{M}{1.5}$ is the time required to get to the maximum speed M, and $(.75\frac{M}{1.5})^2$ is the distance covered during this phase. Similarly, for the deceleration, we have:

$$a=-2\\v_2=-2t+(v_0=M)\\s_2=-t^2+Mt$$ However, since we want the speed where the car stops, we set $v_2=0$ and for time to decelerate, I get: $$t=\frac{M}{2}$$.

Now substituting my solutions for $t$ in terms of $M$ into the distance equation, we can get the total distance spent accelerating: $$s_o=.75t^2=.75(\frac{M}{1.5})^2$$ and the total distance spent decelerating: $$s_2=-(\frac{M}{2})^2+\frac{M^2}{2}$$.

Now I know that $s_0+s_1+s_2=1000$, whereas $s_0$ is time spent accelerating and $s_2$ is time spent slowing down and $s_1$ is time spent at maximum speed: $$s_1=1000-.75(\frac{M}{1.5})^2-\frac{M^2}{2}+\frac{M^2}{4}$$ However, we also know that: $s_1=Mt$. This allows us to compute $t_1$. Knowing that $t_0+t_1+t_3=t_{\mathrm{total}}$, we lead to the conclusion that the total time is: $$t_{\mathrm{total}}=.75(\frac{M}{1.5})^2+(\frac{1}{M})(1000-.75(\frac{M}{1.5})^2-\frac{M^2}{2}+\frac{M^2}{4})+\frac{M}{2}$$. However, this graph produces some interesting results:

Wolfram Alpha Graphed It!

As you can see, there is an infection point where the total time required stops getting smaller and starts getting larger, around 40. However, I computed the maximum speed the car can achieve and still stop in a thousand meters by $s_0+s_2=1000$; substituting $t$ in terms of $M$ and solving yields a maximum speed of about $109 \space \mathrm{m/s}$!

Would someone be so kind as to tell me: what does the inflection point mean, and was my attempt at computing the maximum speed the car can achieve while still being able to stop within a 1000 meters correct?

Thanks!