In physics, Torricelli's equation, or Torricelli's formula, is an equation created by Evangelista Torricelli to find the final velocity of a moving object with constant acceleration along an axis (for example, the x axis) without having a known time interval.
The equation itself is:[1]
where
- is the object's final velocity along the x axis on which the acceleration is constant.
- is the object's initial velocity along the x axis.
- is the object's acceleration along the x axis, which is given as a constant.
- is the object's change in position along the x axis, also called displacement.
In this and all subsequent equations in this article, the subscript (as in ) is implied, but is not expressed explicitly for clarity in presenting the equations.
This equation is valid along any axis on which the acceleration is constant.
Without differentials and integration
[edit]
Begin with the following relations for the case of uniform acceleration:
| | (1) |
| | (2) |
Take (1), and multiply both sides with acceleration
| | (3) |
The following rearrangement of the right hand side makes it easier to recognize the coming substitution:
| | (4) |
Use (2) to substitute the product :
| | (5) |
Work out the multiplications:
| | (6) |
The crossterms drop away against each other, leaving only squared terms:
| | (7) |
(7) rearranges to the form of Torricelli's equation as presented at the start of the article:
| | (8) |
Using differentials and integration
[edit]
Begin with the definitions of velocity as the derivative of the position, and acceleration as the derivative of the velocity:
| | (9) |
| | (10) |
Set up integration from initial position to final position
| | (11) |
In accordance with (9) we can substitute with , with corresponding change of limits.
| | (12) |
Here changing the order of and makes it easier to recognize the upcoming substitution.
| | (13) |
In accordance with (10) we can substitute
with , with corresponding change of limits.
| | (14) |
It follows:
| | (15) |
Since the acceleration is constant, we can factor it out of the integration:
| | (16) |
Evaluating the integration:
| | (17) |
| | (18) |
The factor is the displacement :
| | (19) |
| | (20) |
From the work-energy theorem
[edit]
The work-energy theorem states that
which, from Newton's second law of motion, becomes