Boat Vector Diagram with Current for Navigation

Answer

To achieve the least time, the boat must steer into the current so that its resultant velocity vector points directly from X to Y. The displacement vector resultant is the vector $\vec{R} = \vec{v}_{boat} + \vec{v}_{current}$, which must align with the bearing of $065^\circ$.

Explanation

1. Define the Coordinate Systems Map the directions using a compass rose. Harbour Y is at a bearing of $065^\circ$ from X. The current is moving South-East, which corresponds to a bearing of $135^\circ$ (since East is $090^\circ$ and South is $180^\circ$, halfway is $135^\circ$).

2. Establish the Vector Triangle Let $\vec{v}_{c}$ be the velocity of the current ($1.5 \text{ m/s}$ at $135^\circ$) and $\vec{v}_{b}$ be the velocity of the boat ($5.5 \text{ m/s}$ at an unknown angle $\theta$). The resultant velocity $\vec{v}_{R}$ must lie along the track $065^\circ$. The relationship is defined by: $\vec{v}_{R} = \vec{v}_{b} + \vec{v}_{c}$ (The resultant of the boat and the current equals the total velocity.)

3. Sketching the Components

   • Draw a line representing the track from X to Y at a bearing of $065^\circ$.
   • From X, draw the current vector $\vec{v}_{c}$ pointing towards the South-East ($135^\circ$).
   • From the tip of $\vec{v}_{c}$, draw the boat vector $\vec{v}_{b}$ such that it terminates on the $065^\circ$ line.
   • This forms a triangle where the resultant vector $\vec{v}_{R}$ is the straight line from X to Y.

4. Conceptual Representation

   Vector Component	Magnitude	Direction (Bearing)
   Current ($\vec{v}_{c}$)	$1.5 \text{ m/s}$	$135^\circ$
   Boat ($\vec{v}_{b}$)	$5.5 \text{ m/s}$	$\theta$ (variable)
   Resultant ($\vec{v}_{R}$)	$	\vec{v}_{R}

Final Answer

The resultant vector lies on the line connecting X and Y, with a bearing of $065^\circ$, constructed by the vector addition of the boat's velocity and the current's drift. $\vec{v}_{R} = \vec{v}_{b} + \vec{v}_{c}$ The resultant velocity is the vector sum of the boat's steering velocity and the current's flow velocity.

Common Mistakes

• Confusing Relative and Resultant Velocity: Students often draw the boat vector towards Y directly, forgetting that the current will push the boat off course unless the boat compensates by aiming into the current.
• Bearing Errors: Miscalculating South-East ($135^\circ$) or misinterpreting the direction of the current relative to the bearing of the harbour. Remember that $000^\circ$ is North and bearings increase clockwise.