If the radius of a pipe carrying water is doubled, how is the velocity of the water affected?

To understand how the velocity of water is affected when the radius of a pipe is doubled, we can use the principle of conservation of mass, specifically the equation of continuity for incompressible fluids. The equation states that the product of the cross-sectional area (A) and the velocity (v) of the fluid must remain constant along the pipe if there is no influx or outflux of fluid. This can be expressed as:

A1 * v1 = A2 * v2

Where A1 and v1 are the cross-sectional area and velocity at one point in the pipe, and A2 and v2 are the corresponding values at another point.

The cross-sectional area of a circular pipe is calculated using the formula A = πr², where r is the radius. If the radius of the pipe is doubled, the new area (A2) becomes:

A2 = π(2r)² = π(4r²) = 4πr²

This means the cross-sectional area quadruples when the radius is doubled. According to the conservation of mass, since the overall flow rate must remain constant, if the area increases, the velocity must decrease to compensate.

Thus, if the area increases by a factor of