Physics Problem-drain plug

Question

A rectangular trough, 1.7 m long, 0.60 m wide, and 0.40 m deep, is completely full of water. One end of the trough has a small drain plug right at the bottom edge.

When you pull the plug, at what speed does water emerge from the hole?

Answer 1

Let

h = 0.40 m be the depth of the trough,

ρ be the density of water,

g = 9.81 m/s² be gravitation acceleration,

p = ρgh be the pressure at the bottom of the trough,

v (to be calculated) be the velocity of water coming out.

This problem can be solved by conservation of energy of some

small volume V of water located at the opening.

This volume V of water has mass

m = Vρ

Before exiting the trough, that volume V of water has potential pressure energy

Vp = Vρgh

This energy gets converted to kinetic energy upon exit, whose magnitude is

mv²/2 = Vρv²/2

By conservation of energy

Vρv²/2 = Vρgh

v = √(2gh)

Substituting actual numbers,

v = √(2×9.81×0.40) = 2.8 m/s

Notice that the solution depends only on the depth of the trough, not any of the other dimension.

Also it is independent of ρ, meaning, independent of the liquid.