Boats and streams

Learning outcome

Separate a boat’s speed relative to water from its speed relative to the bank, and solve upstream, downstream and round-trip problems.

Concepts and assumptions

Still-water speed u is the boat’s speed relative to water. Current speed c is the water’s speed relative to the bank. Journey distances and times use the bank as reference.

Assume straight travel along a uniform current, constant still-water speed in both directions, no stops or wind effects, and instantaneous turns.

Downstream, boat and current move together: downstream speed = u + c.

Upstream, the current opposes the boat: upstream speed = u - c.

Require u > c ≥ 0 for upstream progress. At u = c, the boat makes no upstream progress relative to the bank. At u < c, it drifts downstream; a negative difference is not a usable upstream speed.

Let downstream speed be D and upstream speed U. Adding D = u + c and U = u - c cancels current: u = (D + U)/2. Subtracting cancels still-water speed: c = (D - U)/2. Both observations must involve the same boat under unchanged conditions.

Time = distance ÷ the relevant bank-relative speed. For round trips, add both distances and both times. Averaging D and U gives still-water speed, not generally journey average speed: equal distances take longer upstream.

Keep units compatible; convert minutes to hours when using km/h.

Worked examples

Example 1 — Find a journey time. A boat has still-water speed 14 km/h in a current of 3 km/h. Downstream speed = 14 + 3 = 17 km/h; upstream speed = 14 - 3 = 11 km/h. To travel 51 km downstream, time = 51 ÷ 17 = 3 h.

Example 2 — Recover both speeds. The same boat travels 42 km downstream in 2 h and 30 km upstream in 2 h 30 min = 2.5 h. D = 42/2 = 21 km/h; U = 30/2.5 = 12 km/h. Hence u = (21 + 12)/2 = 16.5 km/h and c = (21 - 12)/2 = 4.5 km/h.

Example 3 — Unknown distance. A boat with u = 15 km/h and c = 3 km/h completes equal distances downstream and upstream in 5 h without stops. Speeds are 15 + 3 = 18 km/h and 15 - 3 = 12 km/h. For one-way distance d km, d/18 + d/12 = 5. Thus (2 × d + 3 × d)/36 = 5; 5 × d = 180, so d = 36 km. Total distance = 72 km; average speed = 72/5 = 14.4 km/h, not 15 km/h.

Common mistakes

Do not add current twice or use still-water speed directly for upstream time. Changed boat effort or current invalidates the recovery formulas.

Practice questions

1. Still-water speed is 13 km/h and current speed is 2 km/h. Find both bank-relative speeds and the time for 44 km upstream.
2. A boat’s downstream and upstream speeds are 18 km/h and 10 km/h. Find still-water speed and current speed.
3. Still-water speed is 20 km/h and current speed is 4 km/h. Find the time and average speed for 48 km each way, without stops.
4. A boat with still-water speed 12 km/h covers 36 km upstream in 4 h. Find the current speed and the time for 45 km downstream under unchanged conditions.

Worked answers

1. Downstream speed = 13 + 2 = 15 km/h; upstream speed = 13 - 2 = 11 km/h. Upstream time = 44 ÷ 11 = 4 h.
2. Still-water speed = (18 + 10)/2 = 14 km/h. Current speed = (18 - 10)/2 = 4 km/h.
3. Downstream speed = 20 + 4 = 24 km/h; upstream speed = 20 - 4 = 16 km/h. Times = 48/24 = 2 h and 48/16 = 3 h. Total time = 5 h; average = 96/5 = 19.2 km/h.
4. Upstream speed = 36/4 = 9 km/h. Current = 12 - 9 = 3 km/h. Downstream speed = 12 + 3 = 15 km/h, so time = 45/15 = 3 h.