Standard +0.8 This is a relative velocity problem requiring vector decomposition, finding the closest approach distance using perpendicular distance from a line of motion. It involves multiple steps: setting up coordinates, calculating relative velocity after the turn, finding the perpendicular component, and applying trigonometry. While M4 students should be familiar with relative motion, the 30° turn and geometric setup require careful reasoning beyond routine exercises.
Two ships \(A\) and \(B\) are sailing in the same direction at constant speeds of 12 km h\(^{-1}\) and 16 km h\(^{-1}\) respectively. They are sailing along parallel lines which are 4 km apart. When the distance between the ships is 4 km, \(B\) turns through 30° towards \(A\).
Find the shortest distance between the ships in the subsequent motion. [7]
Two ships $A$ and $B$ are sailing in the same direction at constant speeds of 12 km h$^{-1}$ and 16 km h$^{-1}$ respectively. They are sailing along parallel lines which are 4 km apart. When the distance between the ships is 4 km, $B$ turns through 30° towards $A$.
Find the shortest distance between the ships in the subsequent motion. [7]
\hfill \mbox{\textit{Edexcel M4 2005 Q3 [7]}}