PHYSICS IA HL
Torricelli’s Law: Efflux Velocity
1. INTRODUCTION AND BACKGROUND
Ever since I began the HL Physics course, fluid dynamics have greatly interested me – I remember skimming through the optional engineering
unit and pausing to read the section on it and being utterly captivated. This led to more research on my part, and I spent hours looking up
concepts such as fluid turbulence and Stoke’s Law. In my last investigation, I explored the terminal velocity of a solid within a liquid, and a
piece of the apparatus I used to contain the water had closed spouts in it. Upon further investigation, I discovered that the velocity of a liquid
leaking from the spouts would vary according to the pressure exerted on it. This investigation will examine the relationship between the pressure
on a liquid and its ejection velocity from a spout with use of Torricelli’s Law – how does the velocity of efflux of a liquid change when exposed
to varying volumes of water? This law implies that the speed of efflux of a liquid under the influence of gravity is proportionally joint to the
square root of the vertical distance h between the surface of the liquid and the center of the opening, as well as proportional to the square root of
twice the acceleration caused by gravity. This law is derived from Bernoulli’s principle, and can be expressed as shown below, whereby v1 is
velocity of the water flowing, h is the vertical distance between the liquid surface and the center of the opening height, and g is the acceleration
due to gravity, being 9.81ms-2 for this investigation:
𝑣1 = √2𝑔ℎ
From this, we can determine that the speed of a small amount of water flowing through an open tank at a distance of h is the same as that of a
drop of water free-falling from distance h, ignoring the effects of air-resistance – efflux speed is independent to the direction of its flow. An
easier way to derive this law excluding Bernoulli’s principle is by simply equating kinetic energy to gravitational potential energy, giving us this
form:
1
𝑚𝑣 2 = 𝑚𝑔ℎ
2
𝑣 2 = 2𝑔ℎ
𝑣 = √2𝑔ℎ
1
, PHYSICS IA HL
The predicted trend is that efflux velocity will increase as pressure increases, where there will be a linear relationship between v2 and h, where
the gradient works out to be 2g. This is a largely relevant law to investigate today, due to its applications; Torricelli’s law can be applied in
combination with Bernoulli’s Principle to aid the use of hydro-dams, where water is released to create hydroelectric energy via the spinning of a
turbine, a renewable source. Since the world is merely a decade away from reaching a temperature increase of +2.0C, a point where we cannot
reverse the damage done to our planet, researching and using clean sources of energy is more imperative than ever.
2. APPARATUS
Fig. I
Equipment Quantity Uncertainty Purpose
675ml cylinder 1 N/A Container for water
with spouts
Water 1 ±0.5ml Liquid for calculations of
efflux velocity
250ml beaker 1 ±0.5ml For measurements of water
levels
Video Camera 1 ± 0.1s To record the water efflux
displacement
Ruler, 30cm 1 ±0.01m To measure value of h
Ruler, 1m 1 ±0.01m To measure horizontal
displacement of efflux (s)
Blue Tack 2 pieces N/A To block up unwanted
spouts and prevent them
from leaking water
2
Torricelli’s Law: Efflux Velocity
1. INTRODUCTION AND BACKGROUND
Ever since I began the HL Physics course, fluid dynamics have greatly interested me – I remember skimming through the optional engineering
unit and pausing to read the section on it and being utterly captivated. This led to more research on my part, and I spent hours looking up
concepts such as fluid turbulence and Stoke’s Law. In my last investigation, I explored the terminal velocity of a solid within a liquid, and a
piece of the apparatus I used to contain the water had closed spouts in it. Upon further investigation, I discovered that the velocity of a liquid
leaking from the spouts would vary according to the pressure exerted on it. This investigation will examine the relationship between the pressure
on a liquid and its ejection velocity from a spout with use of Torricelli’s Law – how does the velocity of efflux of a liquid change when exposed
to varying volumes of water? This law implies that the speed of efflux of a liquid under the influence of gravity is proportionally joint to the
square root of the vertical distance h between the surface of the liquid and the center of the opening, as well as proportional to the square root of
twice the acceleration caused by gravity. This law is derived from Bernoulli’s principle, and can be expressed as shown below, whereby v1 is
velocity of the water flowing, h is the vertical distance between the liquid surface and the center of the opening height, and g is the acceleration
due to gravity, being 9.81ms-2 for this investigation:
𝑣1 = √2𝑔ℎ
From this, we can determine that the speed of a small amount of water flowing through an open tank at a distance of h is the same as that of a
drop of water free-falling from distance h, ignoring the effects of air-resistance – efflux speed is independent to the direction of its flow. An
easier way to derive this law excluding Bernoulli’s principle is by simply equating kinetic energy to gravitational potential energy, giving us this
form:
1
𝑚𝑣 2 = 𝑚𝑔ℎ
2
𝑣 2 = 2𝑔ℎ
𝑣 = √2𝑔ℎ
1
, PHYSICS IA HL
The predicted trend is that efflux velocity will increase as pressure increases, where there will be a linear relationship between v2 and h, where
the gradient works out to be 2g. This is a largely relevant law to investigate today, due to its applications; Torricelli’s law can be applied in
combination with Bernoulli’s Principle to aid the use of hydro-dams, where water is released to create hydroelectric energy via the spinning of a
turbine, a renewable source. Since the world is merely a decade away from reaching a temperature increase of +2.0C, a point where we cannot
reverse the damage done to our planet, researching and using clean sources of energy is more imperative than ever.
2. APPARATUS
Fig. I
Equipment Quantity Uncertainty Purpose
675ml cylinder 1 N/A Container for water
with spouts
Water 1 ±0.5ml Liquid for calculations of
efflux velocity
250ml beaker 1 ±0.5ml For measurements of water
levels
Video Camera 1 ± 0.1s To record the water efflux
displacement
Ruler, 30cm 1 ±0.01m To measure value of h
Ruler, 1m 1 ±0.01m To measure horizontal
displacement of efflux (s)
Blue Tack 2 pieces N/A To block up unwanted
spouts and prevent them
from leaking water
2
