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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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
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