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Question

Two cars A and B are moving west to east and south to north respectively along crossroads. A moves with a speed of 18 kmh1 and is 125 m away from the point of intersection of crossroads and B moves with a speed of 13.5 kmh1 and is 100 m away from the point of intersection of crossroads. Find the shortest distance between them (in m)

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Solution

Converting velocities in SI units,
Velocity of A =18×518=5 m/s
Velocity of B =13.5×518=3.75 m/s


After time t, let us plot the components of velocity of A and B in the direction along A'B'. (Where αf is the angle made by the line A'B' with the x-axis)

When the distance between the two is minimum, the relative velocity of approach is zero.

5cosαf+(3.75sinαf)=0
5cosαf=3.75sinαf
tanαf=53.75=43
Distance of particle B from the origin after time t=1003.75t
Distance of particle A from the origin after time t=1255t
tanαf=1003.75t1255t=43
t=1285 s
So, OB'=4 m and OA'=3 m
A'B'=42+(3)2=5 m

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