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Work & motion

Boats & streams: separate boat and current

Understand how current changes a boat's speed over the ground and recover still-water speed from a round trip.

IntermediateAbout 12 min3 worked examples
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01 / THE STARTING POINT

Before you begin

What you need: Relative speed and simple equations.

  • Calculate upstream and downstream speeds
  • Recover boat and stream speeds
  • Find journey times in both directions
02 / BUILD THE CONCEPT

Understand the idea

The boat moves relative to the water

Still-water speed b describes the boat's motion through the water. The stream adds its own speed s relative to the bank. Standard questions assume both remain constant.

Direction changes the effect of current

Downstream, the current assists: speed = b + s. Upstream, it opposes: speed = b − s. To make upstream progress, b must exceed s.

Add and subtract to recover the parts

If downstream speed is d and upstream speed is u, adding gives d + u = 2b; subtracting gives d − u = 2s. This separates boat and current contributions.

03 / A DIFFERENT WAY TO SEE IT

The current helps one way, resists the other

  1. Boat: 10 km/h
  2. Current: 2 km/h
  3. Downstream: 12 km/h
  4. Upstream: 8 km/h
The boat's still-water speed stays the same. Its speed measured from the bank changes with direction.
04 / FROM IDEA TO ANSWER

A method you can reuse

KEEP THIS HANDY

Downstream = b + s; upstream = b − s

Still-water speed b = (d + u)/2; stream speed s = (d − u)/2.
  1. Identify whether each given speed is still-water, upstream or downstream.
  2. Add or subtract stream speed to obtain speed relative to the bank.
  3. Use distance divided by the relevant speed for each journey stage.
A useful insight

For equal upstream and downstream distances, the upstream leg always takes longer when the current is positive and b > s.

05 / WATCH THE METHOD WORK

Worked examples

Read the question first. Try a step yourself, then compare your reasoning.

EXAMPLE 01

Find both speeds

A boat travels at 10 km/h in still water; current is 2 km/h.

  1. Downstream = 10 + 2.
  2. Upstream = 10 − 2.
  3. Both are speeds relative to the bank.
Answer12 km/h downstream; 8 km/h upstream
EXAMPLE 02

Recover the current

Downstream speed is 18 km/h; upstream speed is 12 km/h.

  1. Boat speed = (18 + 12)/2 = 15.
  2. Stream speed = (18 − 12)/2 = 3.
  3. Check: 15 + 3 = 18 and 15 − 3 = 12.
AnswerBoat 15 km/h; stream 3 km/h
EXAMPLE 03

Return journey

A boat with still-water speed 10 km/h travels 24 km downstream and back in a 2 km/h stream. Find total time.

  1. Downstream time = 24/12 = 2 hours.
  2. Upstream time = 24/8 = 3 hours.
  3. Add the stage times.
Answer5 hours
06 / YOUR TURN

Check your understanding

Quick questions, with explanations. These are for self-study and do not affect your account score.

0 of 3 questions checked

QUESTION 1Boat speed 9 km/h, current 3 km/h. Upstream speed?

See the explanation

6 km/h. Upstream speed = 9 − 3 = 6 km/h.

QUESTION 2Downstream 14 km/h, upstream 10 km/h. Stream speed?

See the explanation

2 km/h. Stream = (14 − 10)/2 = 2 km/h.

QUESTION 3What happens upstream if boat speed equals current speed?

See the explanation

No progress relative to the bank. b − s = 0, so there is no upstream progress in this model.

07 / TAKE THE NEXT STEP

Put your understanding to work

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EXPLORE THE NUMBERS

Boats and Streams Calculator

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Boats and Streams Practice Quiz 1

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Boats and Streams Practice Quiz 2

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Boats and Streams Practice Quiz 3

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