Water in a large river estuary flows at a constant 5.3 km/h on a bearing 165°

Expert intro: This is a vector-and-bearing motion problem. The key idea is to treat the boat’s motion through the water and the water current as two velocity vectors, then add them to get the ground velocity from P to Q.

Detailed walkthrough

Part (a): Sketch the velocity diagram

Represent the motion with a vector triangle:

1. Draw the water velocity vector with magnitude 5.3 km/h on a bearing of 165°.
2. Draw the boat’s velocity through the water vector with magnitude 11.6 km/h in the direction the boat steers.
3. The resultant velocity is the boat’s actual motion over the ground, from P to Q.
4. Mark the angle between the resultant and the water velocity inside the vector triangle.

A clear sketch should show:

• one vector for the current,
• one vector for the boat relative to the water,
• one resultant vector joining the start point to the end point.

Useful structure for the diagram

• Label the current vector as 165°.
• Label the line of travel from P to Q as the resultant.
• Indicate the angle between the current vector and the resultant, since this is the angle needed for vector resolution later.

What the marks are looking for

• a correct vector triangle,
• correct direction labels/bearings,
• a marked angle between the resultant and the water velocity.

Because this part asks for a sketch only, no numerical calculation is required here.

💡 Pitfall guide

A common mistake is to draw the boat’s path and the current as if they were the same thing. They are not: the current is one vector, the boat’s steering direction is another, and the actual path over the ground is the resultant. Another frequent error is to label the 165° bearing as an ordinary interior angle without converting from bearing direction to a proper vector diagram.

🔄 Real-world variant

If the current speed or bearing changed, the same method still works: draw the current vector first, then add the boat’s steering vector head-to-tail, and the resultant gives the true path. If the boat were asked to travel from a different starting point or to a different destination, only the bearing and triangle geometry would change, not the vector-addition method.

🔍 Related terms

vector addition, bearing, resultant velocity