This would be my first question on Physics.stackexchange. As I looked closely if I would not double my question, I try dare to ask the question.

I am doing some calculations together with my kids. They had two physics questions on school which I find particularly interesting.

First question: Calculate the total stopping distance for a car knowing the initial speed v in km/h, the friction $\mu$ and the thinking time t of 1 second.

I solved this one pretty easy using the formula: $$ d = (v * t) + \dfrac{v^{2}}{2*\mu*g} $$

So given the example that the car is travelling with a speed of 144 km/h, assuming g of 9.81 and a friction of 0.3 this will yield:

$$ d = ( 40 * 1) + \dfrac{1600}{2*0.3*9.81} => 311.83146449201496 $$

The second question however is where I made up a mistake in my math. This question is, given a known stopping distance d and a friction $\mu$ and still assuming a thinking time of 1 second, what is the initial speed in km/h?

By putting this initial speed and $\mu$ into the first equation of the first question, the same total stopping distance should be proven.

What would be the formula for that last question?

I am assuming that using: $$ v = \sqrt{2 * \mu\ * g\ * d } $$

but that is not including the thinking time right? Hence my outcome is not correct.