# Torricelli’s equation

Torricelli’s equation is extremely important in the study of uniformly varied motion because it does not depend on time.

**Torricelli’s equation**is extremely important because it is the only one to relate

**traveled space, velocity and acceleration**of a mobile

**without depending on time**. This equation takes its name from the Italian physicist Evangelista Torricelli , responsible for important inventions and scientific discoveries in the 17th century.

**Demonstration of Torricelli’s equation**

Starting from the time function of speed, we have:

**v = v _{0} + at**

Squaring both sides of the equality:

v ^{2} = (v _{0} + at) ^{2}

Now we will develop the remarkable product (v _{0} + at) ^{2} , so that:

v ^{2} = v _{0 }^{2} + 2.v _{0} .at + a ^{2} t ^{2}

For the last two terms of the function, we will isolate the factor 2a:

v ^{2} = v _{0 }^{2} + 2a (v _{0} .t + ½ at ^{2} )

**Equation A**

With equation A in hand, we will proceed to the clockwise function of the position in uniformly varied motion:

S = S _{0} + v _{0} .t + ½ at ^{2}

S – S _{0} = v _{0} .t + ½ at ^{2}

As S – S _{0} = ΔS, we have:

ΔS = v _{0} .t + ½ at ^{2}

**Equation B**

Finally, we will substitute equation B into equation A:

v ^{2} = v _{0 }^{2} + 2a (v _{0} .t + ½ at ^{2} )

**v ^{2} = v _{0 }^{2} + 2aΔS**

**Torricelli’s equation**

Note that **there is no time dependence,** since the terms of the equation are the final velocity of the mobile (v), initial speed of the mobile (v _{0} ), acceleration (a) and the space covered (ΔS).

**Torricelli’s name “Equation” is not adequate!**

The term equation is not correct because what we have above is a correspondence between elements, and not an equality satisfied by just a few values. Therefore, we should call the above term **a Torricelli function** , but traditionally, even if it is not the proper form, this expression is recognized as a Torricelli equation.

**Numerical example**

A car starting from rest has a constant acceleration equal to 5 m/s ^{2} . Determine the distance covered by the mobile when its speed is equal to 72 km/h.

**Resolution:**

Taking the data from the question, we have:

a = 5 m/s ^{2}

v = 72 km/h ÷ 3.6 = 20 m/s

v _{0} = 0 (Mobile starting from rest)

ΔS = ?

Substituting the above values into **Torricelli’s equation** , we have:

v ^{2} = v _{0 }^{2} + 2aΔS

20 ^{2} = 0 + 2.5.ΔS

400 = 10.ΔS

ΔS = 40 m