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

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