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Two particles start simultaneously from the same point and move along two straight lines, one with uniform velocity u and the other from rest with uniform acceleration 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     AIEEE  Solved  Paper-2003

A)                         \[\frac{u\sin \alpha }{f}\]                            

B) \[\frac{f\cos \alpha }{u}\]                           

C) \[u\sin \alpha \]                              

D) \[\frac{u\cos \alpha }{f}\]

Correct Answer: D

Solution :

We have,                 \[{{R}^{2}}={{u}^{2}}+{{f}^{2}}{{t}^{2}}+2uft\cos \,({{180}^{o}}-\alpha )\]                 \[{{R}^{2}}={{u}^{2}}+{{f}^{2}}{{t}^{2}}-2uft\cos -\alpha \] Let          \[V={{u}^{2}}+{{f}^{2}}{{t}^{2}}-2uft\cos \alpha \]                                 \[\frac{dV}{dt}=0=2{{t}^{2}}t-2utf\cos \alpha \]                 \[\frac{{{d}^{2}}V}{d{{t}^{2}}}=2{{t}^{2}}=+ve\] i.e., velocity will be least after a time.                 \[\frac{dV}{dt}=0=2{{f}^{2}}t-2uf\cos \alpha \] \[\Rightarrow 2{{f}^{2}}t=2uf\cos \alpha \] \[\Rightarrow \]               \[t=\frac{2uf\cos \alpha }{2{{f}^{2}}}\] \[\Rightarrow \]               \[t=\frac{u\cos \alpha }{f}\]