277. Flow of Water Through a Pipe · Volume = Area × Speed
The volume of water flowing in unit time always equals the cross-sectional area multiplied by the speed.
What this lesson covers
Try to break it
Drag R to change the pipe's radius and V to change the flow speed. Volume per second always equals (cross-section area) × (speed) = πr² × v. Halve r and the flow drops by a factor of 4 (because area depends on r²); double v and the volume per second doubles.
How you build it
Build the circular cross-section and the flow length, then the volume block: Volume = A × v.
- Point tool: mark O, the centre of the pipe's cross-section.
- Circle tool: click O, then drag to set the radius r — the circular cross-section (area A = πr²).
- Point tool: mark R on the circle — a point on the edge.
- Segment tool: join O to R — the radius r, which fixes the cross-section area.
- Segment tool: from the circle's right edge, draw a horizontal line of length v — how far the water travels in one second (the speed v).
- Polygon tool: trace the volume block — the cross-section pushed along v (height 2r, length v). The water delivered per second is Volume = A × v.
The proof, step by step
Prove that the volume of water flowing per unit time equals cross-sectional area × speed.
- Area of the circular cross-section = πr²
- Length of the water column flowing in 1 second = v (speed)
- Volume of a block = Base Area × Length
- Therefore, Volume flow rate = πr² × v
Worked example
Water flows through a cylindrical pipe of radius 7 cm at a speed of 10 cm/s. Find the volume of water flowing through the pipe in one second. (Take π = 22/7)
Area = πr² = (22/7) × 7² = 154 cm². Volume flow rate = Area × Speed = 154 × 10 = 1540 cm³/s.
- 1540 cm³ — correct
- 1400 cm³
- 154 cm³
- 140 cm³