Particle $A$ moves uniformly with velocity $v$ so that vector $v$ is continually aimed at point $B$ which moves rectilinearly with a velocity $u < v$ . At $t = 0$ , $v$ and $u$ are perpendicular. The time when they converge is given as $t = \frac{x l v}{v^{2} - u^{2}}$ . Find $x$ . Assume $A$ and $B$ are separated by $l$ at $t = 0$

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Solution

$\int_{0}^{t} (v c o s α - u) d τ = 0 (r e l a t i v e d i s p l a c e m e n t a l o n g x a x i s i s z e r o) \Rightarrow v \int_{0}^{t} c o s α d τ - u t = 0 \Rightarrow \int_{0}^{t} c o s α d τ = \frac{u t}{v} (s i n c e u a n d v a r e c o n s t a n t s) \int_{0}^{t} (v - u c o s α) d τ = l (i n t e g r a t i n g v e l o c i t y o f a p p r o a c h g i v e s t h e r e l a t i v e d i s p l a c e m e n t a l o n g t h e l i n e o f a p p r o a c h) \Rightarrow v t - u \int_{0}^{t} c o s α d τ = l \Rightarrow v t - u (\frac{u t}{v}) = l \Rightarrow t = \frac{v l}{v^{2} - u^{2}} = \frac{x v l}{v^{2} - u^{2}} \Rightarrow x = 1$

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