# Minimum time required to reach a point [closed]

A police inspecter P is at point $(\acute{x},\acute{y})$ and a thief X is at point $(x, y)$. X has a constant velocity $V_x\hat{i} + V_y\hat{j}$, where $\hat{i}$ and $\hat{j}$ are unit vectors in $X$ and $Y$ direction respectively. Maximum speed of P is S. What is the minimum time in which P will catch X? (assuming thatthe acceleration and deceleration of P are instantaneous)

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## closed as too localized by Emilio Pisanty, Manishearth♦, Qmechanic♦Dec 10 '12 at 10:17

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Pure homework questions are discouraged by the FAQ. Provide details of the concepts you are struggling with and a reasonable try at the problem, and we can then help you. –  Emilio Pisanty Nov 22 '12 at 17:06
@EmilioPisanty, it is not a homework question. Just a part of coding question on spoj. In this case I knew the concepts, but was unable to find a solution as was taking a very lengthy approach –  Anubhav Agarwal Nov 22 '12 at 18:20

Initial coordinates of the thief: $(x_0, y_0)$, initial coordinates of the officer: $(0,0)$. Assume the thief to have velocity $(t_x,t_y)$. The officer has speed $(p_x,p_y)$ with maximum speed $p_m^2 \geq p_x^2 + p_y^2$.
Since the thief has a constant velocity (i.e. does not run away from the officer), it appears logical that the best way for the officer to catch the thief is to run in a straight towards a particular point along the thief’s trajectory. Hence we have $p_x = \textrm{ const } = p_y$. This has the nice side effect that we don’t have to differentiate anything to find the minimum time $t$ at which the two meet, since they will only meet once. One can then set up equations as following:
$$p_x t = x_0 + t_x t \qquad (1)$$ $$p_y t = y_0 + t_y t \qquad (2)$$ $$p_m^2 = p_x^2 + p_y^2 \qquad (3)$$
Solving $(3)$ for $p_x$, substituting in $(1)$, solving for $p_y$ and then substituting in $(2)$ yields a single equation with only one unknown, $t$. Solving and substituting back into $(1)$ and $(2)$ gives $p_{x,y}$.