An ideal fluid flows through a long horizontal circular pipe. In one region of the pipe, it has radius R. The pipe then widens to radius 2R. What is the ratio of the fluid’s speed in the region of radius R to the speed of the fluid in region with radius 2R?
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The principle of continuity states that for an incompressible fluid, the flow rate (volume per time) is constant. This is given by the equation A₁v₁ = A₂v₂, where A is the cross-sectional area and v is the fluid speed.
The cross-sectional area of a pipe is A = πr². Therefore, the equation becomes:
Simplifying, we get R²v₁ = 4R²v₂. Canceling R² gives v₁ = 4v₂. The ratio of speeds v₁/v₂ is therefore 4.
Final Answer: 4