# Two particles are moving along two long straight lines, in the same plane, with the same speed: 20 cm/s. The angle between the two lines is 60° and their intersection point is O. At a certain...

Two particles are moving along two long straight lines, in the same plane, with the same speed: 20 cm/s. The angle between the two lines is 60° and their intersection point is O. At a certain moment, the two particles are located at distances 3 m and 4 m from O and are moving towards O. Subsequesntly the shortest distance between them will be:

A) 50 cm

B) 40 √2 cm

C) 50 √2 cm

D) 50 √3 cm

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### 1 Answer

Two particles are moving along straight lines, in the same plane. Their speed is the same and equal to 20 cm/s. The angle between the two lines is 60 degrees and the intersect at point O.

At a certain moment, the two particles are located at distances 3 m and 4 m from O and are moving towards O.

The displacement D of the particle that is initially at a distance 3 m from O can be divided into two components D*cos 60 is in the same direction as the other particle and D*sin 60 is perpendicular to the direction in which the particle is moving.

The distance between the particles after t seconds is `S = sqrt((150*sqrt3 - 10*sqrt 3*t)^2 + (250 - 10t)^2)`

The minimum distance between them is at t when S' = 0

=> `(20*t-350)/sqrt(t^2-35*t+325) = 0`

=> t = 17.5

At t = 17.5, S = `50*sqrt 3`

S'' = `(375*sqrt(t^2-35*t+325))/(t^4-70*t^3+1875*t^2-22750*t+105625) `

At t = 17.5, S'' is positive

As S' = 0 and S'' is positive at t = 17.5 this is a point of minimum value.

**The minimum distance between the two particles as `50*sqrt 3` cm**