Speed in Still Water: Formulas, Rules and Examples

Speed in Still Water is the speed of a boat when the water has no current. In a stream, the current changes the boat’s effective speed: it increases downstream and decreases upstream. This page covers the boat speed formula, relations between still-water, upstream and downstream speeds, time calculations and useful boats-and-streams shortcuts.

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What Is Speed in Still Water?

Speed in still water is the natural speed of a boat when there is no water current. If the boat’s still-water speed is b and the stream speed is s, its downstream speed is b + s and its upstream speed is b − s.

Downstream motion is in the direction of the current, so the speeds add. Upstream motion is against the current, so the stream speed is subtracted. For example, if a boat moves at 12 km/h in still water and the stream flows at 3 km/h, its downstream speed is 15 km/h and its upstream speed is 9 km/h.

Speed in Still Water Formula & Tricks

Important Formulas

Downstream speed
Downstream speed = b + s

Here, b is the boat’s speed in still water and s is the speed of the stream.

Upstream speed
Upstream speed = b − s

The current opposes the boat’s motion, so the stream speed is subtracted from the still-water speed.

Speed in still water from two effective speeds
b = (Downstream speed + Upstream speed) / 2

The average of the downstream and upstream speeds gives the boat’s speed in still water.

Stream speed from two effective speeds
s = (Downstream speed − Upstream speed) / 2

Half the difference between downstream and upstream speeds gives the stream speed.

Time, distance and speed
Time = Distance / Speed

Use the effective speed: downstream speed for motion with the current and upstream speed for motion against the current.

Quick Tricks

Use half-sum and half-difference

When downstream and upstream speeds are given, add them and divide by 2 to get still-water speed. Subtract upstream speed from downstream speed and divide by 2 to get stream speed.

Example: If downstream speed is 18 km/h and upstream speed is 10 km/h, still-water speed = (18 + 10) / 2 = 14 km/h, and stream speed = (18 − 10) / 2 = 4 km/h.
Check the direction before applying the formula

Add the stream speed for downstream travel and subtract it for upstream travel. A boat’s upstream speed must be less than its downstream speed when the current is positive.

Example: For b = 20 km/h and s = 5 km/h, downstream speed is 25 km/h, while upstream speed is 15 km/h.
Convert units before calculating time

Use consistent units. For example, convert metres to kilometres or hours to seconds before applying Time = Distance / Speed.

Example: A speed of 36 km/h equals 10 m/s because 36 × 5/18 = 10.

Speed in Still Water Concepts

Downstream and Upstream Speeds

Downstream speed is b + s, while upstream speed is b − s.

A boat moving in the direction of the stream gets additional speed from the current. A boat moving against the current loses speed because the current opposes its motion. If b = 16 km/h and s = 4 km/h, the downstream speed is 20 km/h and the upstream speed is 12 km/h.

Example: A boat travels 30 km downstream at 15 km/h. Its time is 30/15 = 2 hours.

Finding Boat Speed and Stream Speed

The boat’s speed in still water is half the sum of its downstream and upstream speeds, and the stream speed is half their difference.

If D represents downstream speed and U represents upstream speed, then b = (D + U)/2 and s = (D − U)/2. These formulas work because D = b + s and U = b − s.

Example: For D = 22 km/h and U = 14 km/h, b = (22 + 14)/2 = 18 km/h and s = (22 − 14)/2 = 4 km/h.

Time Taken in Upstream and Downstream Travel

Time is calculated using the effective speed in the direction of travel: t = d/(b + s) downstream and t = d/(b − s) upstream.

For the same distance, upstream travel takes more time because b − s is smaller than b + s. If a boat covers 48 km downstream at 24 km/h, it takes 2 hours. At an upstream speed of 16 km/h, the same distance takes 3 hours.

Example: For a 48 km journey with b = 20 km/h and s = 4 km/h, downstream time = 48/24 = 2 hours, while upstream time = 48/16 = 3 hours.

Equal-Distance Upstream and Downstream Journeys

For equal distances, the average speed of a complete upstream-downstream journey is the total distance divided by total time, not the simple average of the two speeds.

If each part has distance d, total distance is 2d and total time is d/(b + s) + d/(b − s). Therefore, average speed = 2(b² − s²)/2b = (b² − s²)/b. This is less than the arithmetic mean b because more time is spent travelling upstream.

Example: If b = 10 km/h and s = 2 km/h, the equal-distance average speed is (10² − 2²)/10 = 96/10 = 9.6 km/h.

Speed in Still Water Video Lessons

Watch short topic-wise lessons for quick revision.

11 Lessons
Lesson 1 of 11 Quick Revision

Boats: Still-Water and Current Speed

Learn how to find a boat’s speed in still water and the speed of the current using upstream and downstream speed relationships.

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Practice Speed in Still Water Questions

Practise published questions related to this topic.

1Which of the following numbers is divisible by 11?→ 2The marked price of an article is 120 percent of its cost price. Its selling price is 90 percent of the marked price. What is the profit percentage?→ 3An article is sold at a profit of 20%. If the selling price is reduced by 10%, what will be the profit percentage?→ 4What is the number of natural numbers less than or equal to 1000 and divisible by 5 or 7?→ 5The marked price of a mobile is ₹10,000. A customer is offered two successive discounts of 20% and 10%.What is the total discount received?→ 6If x = √(1671 + √(93 + √49)) , then what is the value of x?→ 7After two successive discounts of 20 percent and 30 percent an article is sold for Rs. 4480. What is the marked price of the article?→ 8Find the single discount equivalent to two successive discounts of 5% and 8%.→ 9If a shopkeeper sells an article for Rs. 600 and earns 20% profit, what was its cost price?→ 10A person sells 50 chairs for ₹10,000, suffering a loss of 15%. Estimate how many chairs should he sell for ₹9000 to make a profit of 10%?→

Speed in Still Water Quick Quiz

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Quick Revision Notes

Speed in Still Water: Quick Revision

Use these relations to solve boat and stream speed and time questions.

  • Downstream speed = speed in still water + stream speed.
  • Upstream speed = speed in still water − stream speed.
  • Speed in still water = (downstream speed + upstream speed)/2.
  • Stream speed = (downstream speed − upstream speed)/2.
  • Time = distance/speed, using downstream or upstream speed according to the direction.
  • For equal upstream and downstream distances, average speed = 2DU/(D + U) = (b² − s²)/b.
  • The upstream speed must be positive for the boat to move against the current, so b > s.

Speed in Still Water FAQs

What is the still water speed formula when upstream and downstream speeds are known?

Speed in still water = (Downstream speed + Upstream speed) / 2. For speeds 20 km/h and 12 km/h, it is (20 + 12)/2 = 16 km/h.

What is the boat speed formula for the speed of the stream?

Stream speed = (Downstream speed − Upstream speed) / 2. If the two speeds are 20 km/h and 12 km/h, stream speed = (20 − 12)/2 = 4 km/h.

A boat’s speed in still water is 15 km/h and the stream speed is 3 km/h. What are its upstream and downstream speeds?

Downstream speed = 15 + 3 = 18 km/h. Upstream speed = 15 − 3 = 12 km/h.

How much time does a boat take to travel 42 km upstream at 14 km/h?

Time = Distance/Speed = 42/14 = 3 hours.

Why is the average speed for equal upstream and downstream distances not the average of the two speeds?

The time taken in each direction is different. For equal distances, average speed is total distance divided by total time, or 2DU/(D + U), not (D + U)/2.

Can a boat move upstream if the stream speed is greater than its still-water speed?

No. Its upstream effective speed would be b − s, which is zero or negative when s is greater than or equal to b. Upstream movement requires b > s.

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