Mechanics

# Torricelli’s equation

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

In the study of uniformly varied motion , 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:

2 = (v 0 + at) 2

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

2 = v 2 + 2.v 0 .at + a 2 t 2

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

2 = v 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:

2 = v 2 + 2a (v 0 .t + ½ at 2 )

2 = v 2 + 2aΔS
Torricelli’s equation

Don’t stop now… There’s more after the publicity 😉

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

0 = 0 (Mobile starting from rest)

ΔS = ?

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

2 = v 2 + 2aΔS

20 2 = 0 + 2.5.ΔS

400 = 10.ΔS

ΔS = 40 m