Question

A boat travels at 15 m/s in a direction 45° east of north for an hour. The boat then turns and travels at 18 m/s in a direction 5° north of east for an hour.

What is the magnitude of the boat’s resultant vector? Round your answer to the nearest whole number.

m/s

What is the direction of the boat’s resultant vector? Round your answer to the nearest whole degree.

° north of east

Answer 1

First convert to distances (each hour = 3600 s):
- d1 = 15 m/s × 3600 s = 54000 m at 45° east of north
- d2 = 18 m/s × 3600 s = 64800 m at 5° north of east

Use east = +x, north = +y:
- d1x = 54000 sin45° = 38183.77 m, d1y = 54000 cos45° = 38183.77 m
- d2x = 64800 cos5° = 64553.42 m, d2y = 64800 sin5° = 5647.69 m

Resultant displacement:
- Rx = 38183.77 + 64553.42 = 102737.19 m
- Ry = 38183.77 + 5647.69 = 43831.46 m
- R = sqrt(Rx^2 + Ry^2) ≈ 111697 m

Average speed (resultant vector magnitude in m/s) = R / total time (2 h = 7200 s):
- magnitude ≈ 111697 / 7200 ≈ 15.5135 m/s → 16 m/s (nearest whole number)

Direction (measured north of east):
- θ = arctan(Ry / Rx) = arctan(43831.46 / 102737.19) ≈ 23° north of east (nearest degree)

Answers:
- Magnitude: 16 m/s
- Direction: 23° north of east