Torricelli’s Theorem (College Board AP® Physics 1: Algebra-Based)

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Dan Mitchell-Garnett

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Torricelli’s theorem

Consider a container filled with fluid
An opening is cut into the container at a given depth below the fluid's surface
- This could be in the side of the container or the base, as pressure only depends on depth below the surface, not orientation
Torricelli's theorem predicts the speed of fluid exiting the hole
- It assumes that the surface and opening are both open to the same environment and therefore have matching pressures

Derived equation

The speed of fluid leaving an opening in a container is described by the equation:

$v = \sqrt{2 g Δ y}$

Where:
- $v$ = the fluid speed leaving the opening, measured in $m / s$
- $g$ = acceleration due to gravity, measured in $m / s^{2}$
- $Δ y$ = distance from the opening to the fluid's surface, measured in $m$
Interestingly, this equation shows that the speed depends only on depth

Container with fluid leaving an opening

A fluid of density ρ is in a container. There is an opening in the side of the container at a depth of Δy. Dotted lines represent the flow of fluid from the surface to the opening. At the surface, the fluid is at pressure P_1 and has speed v_1. At the opening, the fluid is at pressure P_2 and has speed v_2. — **The speed of fluid exiting through the opening is described by Torricelli's theorem**

Derivation

Step 1: Identify the fundamental equations

Recall Bernoulli's equation:

$P_{1} + ρ g y_{1} + \frac{1}{2} ρ {v_{1}}^{2} = P_{2} + ρ g y_{2} + \frac{1}{2} ρ {v_{2}}^{2}$

Where:
- $P_{1}$ = pressure at point 1, measured in $Pa$
- $ρ$ = fluid density, measured in $kg / m^{3}$
- $y_{1}$ = height of point 1 above reference level, measured in $m$
- $v_{1}$ = fluid speed at point 1, measured in $m / s$
- $P_{2}$ = pressure at point 2, measured in $Pa$
- $y_{2}$ = height of point 2 above reference level, measured in $m$
- $v_{2}$ = fluid speed at point 2, measured in $m / s$

This compares quantities at the surface (point 1) and the opening (point 2)

Step 2: Apply the specific conditions

At the surface, speed is zero, as no fluid is passing through that point, therefore, it can be inferred that:

$v_{1} = 0 m / s$

The reference height can be defined at the same level as the opening, therefore, it can be inferred that:

$y_{1} = Δ y$

$y_{2} = 0 m$

Assume that both the fluid's surface and the opening are open to the same atmosphere, therefore, it can be inferred that:

$P_{1} = P_{2} = P_{a t m}$

Step 3: Combine the specific conditions

Applying these conditions to Bernoulli's equation gives:

$P_{a t m} + ρ g Δ y + \frac{1}{2} ρ \times 0^{2} = P_{a t m} + ρ g \times 0 + \frac{1}{2} ρ {v_{2}}^{2}$

$ρ g Δ y = \frac{1}{2} ρ {v_{2}}^{2}$

The density terms can be canceled out on each side:

$g Δ y = \frac{1}{2} {v_{2}}^{2}$

Rename $v_{2}$ as $v$ and rearrange for fluid speed at the opening:

$v = \sqrt{2 g Δ y}$

This is Torricelli's theorem

Examiner Tip

Understanding the assumptions behind this theorem is as important as the equation itself.

You may be presented with a situation where the pressures are not equal, for example, and just reproducing the above derivation without thought would not score points.

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