Question
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.
$$\frac{{3v}}{{2a}}$$
B.
$$\frac{{2a}}{{5v}}$$
C.
$$\frac{{5v}}{{3a}}$$
D.
$$\frac{{2a}}{{3v}}$$
Answer :
$$\frac{{2a}}{{3v}}$$
Solution :
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 $$O$$
$$ = v\cos {30^ \circ } = \frac{{\sqrt 3 v}}{2}.$$
And its displacement $$ = \frac{{\frac{a}{2}}}{{\cos {{30}^ \circ }}} = \frac{a}{{\sqrt 3 }}.$$
Time taken to converge the points $$ = \frac{{{\text{displacement}}}}{{{\text{velocity}}}} = \frac{{2a}}{{3v}}.$$
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 $$O$$
$$ = v\cos {30^ \circ } = \frac{{\sqrt 3 v}}{2}.$$
And its displacement $$ = \frac{{\frac{a}{2}}}{{\cos {{30}^ \circ }}} = \frac{a}{{\sqrt 3 }}.$$
Time taken to converge the points $$ = \frac{{{\text{displacement}}}}{{{\text{velocity}}}} = \frac{{2a}}{{3v}}.$$
