Torricelli’s equation

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

The importance of the Torricelli equation comes from the fact that it does not depend on the time interval.
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

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