Two Ships and a Bearing
Ship A leaves port O at noon travelling due east at 12 km/h. Ship B leaves the same port at 13:00 travelling on a bearing of 060° at 16 km/h. A harbour master wants to know how close the two ships come to each other and when that closest approach happens, so she can warn them if they pass within 3 km. She sets t as the number of hours after noon and writes position vectors in a coordinate system with east as i and north as j. She is unsure whether the closest-approach time is before or after B even sets sail, and she is also unsure whether the bearing 060° should be converted using sine and cosine of 60° or of 30°.
1. Write position vectors for A and B as functions of t (hours after noon), valid for t ≥ 1, and form the separation vector.
Read the worked answer
A travels due east at 12 km/h from t = 0: r_A = 12t i.
B leaves at t = 1 on bearing 060°. A bearing is measured clockwise from north, so the direction makes 30° with east. Hence the velocity of B is 16(cos 30° i + sin 30° j) = 16(√3/2 i + ½ j) = (8√3 i + 8 j) km/h. For t ≥ 1, r_B = (t − 1)(8√3 i + 8 j).
The separation is d = r_B − r_A = [8√3(t − 1) − 12t] i + 8(t − 1) j.
Simplified: the i-component is (8√3 − 12)t − 8√3 and the j-component is 8t − 8. The common error is to use cos 60° for the east component; because bearings are from north, the east component uses cos 30°, the north component sin 30°.
2. Find the time of closest approach for t ≥ 1 and the minimum distance, and decide whether the 3 km warning is triggered.
Read the worked answer
Let D(t) = |d|². Using the components above:
d_i = (8√3 − 12)t − 8√3, d_j = 8t − 8.
With √3 ≈ 1.732, 8√3 − 12 ≈ 1.856, and 8√3 ≈ 13.856.
D(t) = (1.856t − 13.856)² + (8t − 8)².
D'(t) = 2(1.856)(1.856t − 13.856) + 2(8)(8t − 8) = 0.
6.89t − 51.43 + 128t − 128 = 0 → 134.89t = 179.43 → t ≈ 1.33 h after noon (about 13:20).
This is after B departs (t ≥ 1), so it is valid. Minimum distance: d_i ≈ 1.856(1.33) − 13.856 ≈ −11.39; d_j ≈ 8(0.33) ≈ 2.64. Then |d| = √(129.7 + 7.0) ≈ √136.7 ≈ 11.7 km.
The closest approach is about 11.7 km, well outside 3 km, so no warning is needed. A useful check: the ships are already far apart at t = 1 (A is 12 km east) and diverge, so a large minimum is plausible.