Physics Problem: Boat Crossing a River with Current

Scenario

• Boat speed relative to water: 10 km/h heading north
• River current: 5 km/h flowing east

Solution: Vector Addition

The boat's velocity relative to ground (v) is the vector sum of its velocity relative to water (v_b) and water's velocity relative to ground (v_w).

v_b = 10 km/h north (0°)
v_w = 5 km/h east (90°)

Magnitude (Speed)

Since vectors are perpendicular (θ = 90°), use Pythagorean theorem:

v = √(v_b² + v_w²) = √(10² + 5²) = √125 ≈ 11.18 km/h

Direction

Angle north of east: tan θ = v_b / v_w = 10/5 = 2 → θ = arctan(2) ≈ 63.4° north of east
(Or 26.6° east of north)

Result

The stationary observer sees the boat moving at 11.18 km/h at 63.4° north of east (or 26.6° east of north). The current pushes it downstream while it crosses.