Speed of Stream Formula: (Downstream − Upstream) / 2

The speed of stream formula is $s = \frac{v_{down} - v_{up}}{2}$ : subtract the boat's upstream speed from its downstream speed and halve the result. It works because the current helps the boat one way and holds it back the other way by the same amount.

The partner result is the boat's speed in still water, $b = \frac{v_{down} + v_{up}}{2}$ . Both come from relative velocity along a line, and they turn up in physics, in aptitude tests and in river crossing problems. The flow of water itself, how fast and how smoothly it moves, is covered on the fluids formula sheet.

Quick facts

Formula
$s = \frac{v_{down} - v_{up}}{2}$
Boat in still water
$b = \frac{v_{down} + v_{up}}{2}$
Downstream speed
$v_{down} = b + s$
Upstream speed
$v_{up} = b - s$
SI unit
m/s; boat problems often use km/h
Dimensional formula
[M⁰ L¹ T⁻¹]

Where the speed of stream formula comes from

Let b be the boat's speed in still water and s the speed of the current. Going with the current, the two add. Going against it, the current subtracts:

$v_{down} = b + s \qquad v_{up} = b - s$

Subtract the second line from the first and the b cancels: $v_{down} - v_{up} = 2s$ , so $s = \frac{v_{down} - v_{up}}{2}$ . Add the lines instead and the s cancels, giving $b = \frac{v_{down} + v_{up}}{2}$ . Two equations, two unknowns, nothing to memorise beyond the first pair.

Symbols used in boat and stream problems
Symbol	Meaning	Unit
s	speed of stream (current)	m/s or km/h
b	speed of boat in still water	m/s or km/h
$v_{down}$	speed with the current	m/s or km/h
$v_{up}$	speed against the current	m/s or km/h

Solved boat and stream problems

Example 1: a round trip

A boat travels 30 km downstream in 2.0 h and returns the same 30 km in 3.0 h. Find the speed of the stream and the boat's speed in still water.

Downstream: $\frac{30}{2.0} = 15\ \text{km/h}$ . Upstream: $\frac{30}{3.0} = 10\ \text{km/h}$ .

$s = \frac{15 - 10}{2} = 2.5\ \text{km/h} \qquad b = \frac{15 + 10}{2} = 12.5\ \text{km/h}$

Example 2: a swimmer, in m/s

A swimmer covers 90 m downstream in 50 s and 90 m upstream in 150 s. Find the speed of the current.

$v_{down} = \frac{90}{50} = 1.8\ \text{m/s}$ , $v_{up} = \frac{90}{150} = 0.60\ \text{m/s}$ .

$s = \frac{1.8 - 0.60}{2} = 0.60\ \text{m/s}$

The swimmer's own speed in still water is (1.8 + 0.60) / 2 = 1.2 m/s.

Example 3: finding the stream from total time

A motorboat that moves at 5.0 km/h in still water goes 12 km downstream and back in 5.0 h altogether. Find the speed of the stream.

$\frac{12}{5 + s} + \frac{12}{5 - s} = 5 \;\Rightarrow\; \frac{120}{25 - s^2} = 5 \;\Rightarrow\; s^2 = 1$

So s = 1.0 km/h. Check: 12/6 + 12/4 = 2 + 3 = 5 h.

Use one unit for every speed; 1 m/s is the same as 3.6 km/h.

Crossing a river

When a boat heads straight across a river of width w, the current carries it downstream while it crosses. The two motions are at right angles and independent.

Time to cross: $t = \frac{w}{b}$ , set only by the boat's own speed.
Drift downstream: $x = s\,t$ .
To land directly opposite, the boat must aim upstream at an angle θ with $\sin\theta = \frac{s}{b}$ , which only works when b is greater than s.

Example: drift across a river

A river is 120 m wide and flows at 1.5 m/s. A boat heads straight across, moving through the water at 4.0 m/s. How long does it take, and how far downstream does it land?

$t = \frac{120}{4.0} = 30\ \text{s}$ , drift $x = 1.5 \times 30 = 45\ \text{m}$ .

Speed, velocity and units

All four quantities are speeds, so the SI unit is m/s, with dimensions [M⁰ L¹ T⁻¹]. Strictly, the boat and current have velocities with directions; along a straight river you can use signed speeds. The speed, distance and time formula page covers the basic link between those three, and the general vector rule on the relative velocity formula page.

Mistakes in boat problems

Forgetting to halve. The difference of the two speeds is twice the stream speed.
Averaging times instead of speeds. Work out each speed from distance and time first.
Mixing km/h and m/s in the same equation.
Thinking the current changes crossing time. Heading straight across, only the boat's speed sets the time.

Frequently asked questions

What is the formula for the speed of the stream?

Speed of stream = (downstream speed − upstream speed) / 2. The boat's speed in still water is (downstream speed + upstream speed) / 2.

How do you find downstream and upstream speed?

Downstream speed = boat speed + stream speed, upstream speed = boat speed − stream speed, with the boat speed measured in still water.

Can the speed of the stream be greater than the boat's speed?

Yes, but then the boat cannot make headway upstream: b − s would be zero or negative, so it is carried back.

Which unit should I use for boat problems?

Any one unit for all speeds. Physics answers use m/s; aptitude questions usually stay in km/h.