question_answer

Three points are located at the vertices of an equilateral triangle whose side equal to a. They all start moving simultaneously with velocity v constant in modulus, with first point heading continually for the second, the second for the third, and the third for the first. How soon will the points converge?        

A) \[3v/2a\]           

B)        \[2a/5v\]

C) \[5v/3a\]           

D)        \[~2a/3v\]

Correct Answer: D

Solution :

[d] The motions of the points are sketched in the figure. As they start moving simultaneously symmetrically; they will meet at the centroid of the triangle.

The velocity of any point towards centroid of triangle O \[\text{=}\,\text{v cos}\,\text{30}{}^\circ =\frac{\sqrt{3}\text{v}}{2}.\]

And its displacement \[=\frac{a/2}{\cos 30{}^\circ }=\frac{a}{\sqrt{3}}.\].

Time taken to coverage the points \[=\frac{\text{displacement}}{\text{velocity}}=\frac{2a}{3v}.\]