I am not sure how to show that the two formulas for the braking distance between two trains is correct in problem 2-46 of RHK. Any suggestions?
To solve problem 2-46, first write down equations for the position of each train as a function of time (let x1 denote the position of train 1 and similiarly let x2 denote the position of train2 ). Having a collision is equivalent to the condition that the two trains be at the same position at the same time. Hence, set x1(t) = x2(t). This will give you a quadratic equation, which you can solve with the quadratic formula. The key point is that you only get a collision if you get a real answer from the quadratic formula, i.e. you only get a collision when the quantity in the square-root is positive.