If one punctures a small hole in the side of a large reservoir of liquid, at a point below the surface level of the liquid, one will have a jet of liquid from the hole. The lower the hole, the faster this jet will be. If the liquid in the reservoir has a depth of *H*, at what height *h* above the bottom of the reservoir should one place the hole so that the jet gets the most range?

In this situation, we can use Torricelli’s law, which says that the speed of efflux from the hole is , where *z* is the distance of the hole below the liquid surface; this is equal to the velocity an object gains from falling freely through a height *z*.

Here, *z*=*H*–*h*, so our fluid is launched horizontally from a height *h* with speed .

Now, when an object is launched with horizontal speed *v* from a height *h*, the kinematic equations for position as a function of time are

.

Eliminating time between the equations, one finds

.

Solving for *y*=0, we get a range of

.

Now, plugging in our velocity, we get

.

This is at its maximum when is maximal, which occurs at . The maximum range is thus , the total depth of the liquid in the reservoir.

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Tags: Friday Physics, Kinematics, physics, Torricelli's Law

This entry was posted on December 11, 2009 at 1:12 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
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January 1, 2010 at 12:07 am |

[…] Friday 102 By twistedone151 Torricelli’s law, mentioned in the previous Physics Friday, is derived from Bernoulli’s Principle, via a few assumptions. Let us have a small hole at […]