Correct Answer - Option 2 : 2.6 hrs
Concept:
The direction of velocity at any point on the path of an object is tangential to the path at that point and is in the direction of motion.
Since, one ship is at origin, the time taken by the two ships to attain minimum distance is obtained by finding the distance between ship A and ship B and by finding velocity difference of ship A and ship B.
Calculation:
The ship A is moving from the origin.
Velocity of ship A \(= 30{\rm{\vec i}} + 50{\rm{\vec j}}\)
Distance travelled by ship A \(= 0{\rm{\vec i}} + 0{\rm{\vec j}}\)
The ship B is moving from a point.
Velocity of ship B = - 10i
(Negative sign is included because it is travelling opposite to the east direction)
Distance travelled by ship B \(= 80{\rm{\vec i}} + 150{\rm{\vec j}}\)
Time taken by the two ships to attain minimum distance \({\rm{T}} = - \frac{{{{{\rm{\vec r}}}_{{\rm{BA}}}}.{{{\rm{\vec v}}}_{{\rm{BA}}}}}}{{{{\left| {{{{\rm{\vec v}}}_{{\rm{BA}}}}} \right|}^2}}}\)
\({{\rm{\vec r}}_{{\rm{BA}}}} = \left( {80{\rm{\vec i}} + 150{\rm{\vec j}}} \right) - \left( {0{\rm{\vec i}} + 0{\rm{\vec j}}} \right) = 80{\rm{\vec i}} + 150{\rm{\vec j}}\)
\({{\rm{\vec v}}_{{\rm{BA}}}} = \left( { - 10{\rm{i}}} \right) - \left( {30{\rm{\vec i}} + 50{\rm{\vec j}}} \right) = - 40{\rm{\vec i}} - 50{\rm{\vec j}}\)
\({\left| {{{{\rm{\vec v}}}_{{\rm{BA}}}}} \right|^2} = {40^2} + {50^2}\)
= 1600 + 2500 = 4100
Now, on substituting the values,
\(\Rightarrow {\rm{T}} = - \frac{{\left( {80{\rm{\vec i}} + 150{\rm{\vec j}}} \right).\left( { - 40{\rm{\vec i}} - 50{\rm{\vec j}}} \right)}}{{4100}}\)
\(\Rightarrow {\rm{T}} = - \frac{{\left( { - 3200 - 7500} \right)}}{{4100}} = \frac{{10700}}{{4100}} = 2.61\)
Thus, A will be at minimum distance from B in 2.6 hours.
