Boats and streams

The essentials

A boat moving with the current goes downstream, and against the current goes upstream. If the speed of the boat in still water is u km/h and the speed of the stream is v km/h (with u > v), then

• downstream speed = u + v,
• upstream speed = u − v.

Conversely, if the downstream speed is a and the upstream speed is b, then u = (a + b)/2 and v = (a − b)/2.

All the usual relations of distance, speed and time apply: time = distance ÷ speed, using the correct effective speed for each direction.

Key points

• Example: a boat of speed 8 km/h in a stream of 2 km/h covers 30 km upstream in 30 ÷ 6 = 5 hours.
• For a round trip of d km each way, total time = d/(u + v) + d/(u − v).
• The same ideas apply to a swimmer in a river or an aircraft with or against the wind.
• Averages of speeds: the average speed for a round trip is NOT the boat's speed in still water; it equals (u² − v²)/u.

Exam tips

• Write the two effective speeds first, then form equations for times or distances.
• If an equation becomes quadratic, reject any root that makes the upstream speed zero or negative.

Common mistakes

• Adding the stream speed when going upstream.
• Taking the still-water speed as the average of downstream and upstream distances instead of speeds.