question_answer

Two particles start simultaneously from the same point and move along two straight lines, one with uniform velocity 'v' and other with uniform acceleration 'a'. If \[\alpha \] is the angle between the lines of motion of two particles, then the least value of relative velocity will be at time given by

A) \[\frac{v\,\sin \alpha }{a}\]         

B) \[\frac{v\,\cos \alpha }{a}\]

C) \[\frac{v\,\tan \alpha }{a}\]        

D) \[\frac{v\,\cot \alpha }{a}\]

Correct Answer: B

Solution :

[b] \[\overrightarrow{{{v}_{1}}}=v\hat{i}\] \[\overrightarrow{{{v}_{2}}}=at\,\cos \,\alpha \hat{i}+at\,\sin \alpha \hat{j}\] \[{{v}_{2\to 1}}=\left( at\,\cos \alpha -v \right)\hat{i}+\left( at\,\sin \alpha  \right)\hat{j}\] \[\left| {{v}_{2\to 1}} \right|=\sqrt{{{\left( at\,\,\cos \,\,\alpha -v \right)}^{2}}+{{\left( at\,\,\sin \alpha  \right)}^{2}}}\] \[=\sqrt{{{a}^{2}}{{t}^{2}}+{{v}^{2}}-2tv\,\cos \alpha }\] We get \[\frac{d{{v}_{rel}}}{dt}=0\] \[t=\frac{v\,\cos \alpha }{a}\]