Question

Two particles start simultaneously from the same point and move along two straight lines, one with uniform velocity $\vec c{ u }$ and the other from rest with uniform acceleration $\vec{ f }$. Let $\alpha$ be the angle between their directions of motion. The relative velocity of the second particle w.r.t. the first is least after a time:

• A. $\frac{u \sin \alpha}{f}$
• B. $\frac{f \cos \alpha}{u}$
• C. $u \sin \alpha$
• D. $\frac{u \cos \alpha}{f}$

Answer 1

Answer: (D)

We have
$R^{2} =u^{2}+f^{2} t^{2}+2 u f t \cos \left(180^{\circ}-\alpha\right) $
$R^{2} =u^{2}+f^{2} t^{2}-2 u f t \cos \alpha $
Let $V =u^{2}+f^{2} t^{2}-2 u f t \cos \alpha$

$\frac{d V}{d t}=0+2 f^{2} t-2 u f \cos \alpha $
$\frac{d^{2} V}{d t^{2}}=2 f^{2}=+ ve$
i. e., velocity will be least after a time.
$\frac{d V}{d t}=0 =2 f^{2} t-2 u f \cos \alpha $
$\Rightarrow 2 f^{2} t =2 u f \cos \alpha $
$\Rightarrow t=\frac{2 u f \cos \alpha}{2 f^{2}} $
$\Rightarrow t=\frac{u \cos \alpha}{f}$