Torricelli's equation

In physics, Torricelli's equation, or Torricelli's formula, is an equation created by Evangelista Torricelli to find the final velocity of an object moving with a constant acceleration along an axis (for example, the x axis) without having a known time interval.

The equation itself is:[1]

\( {\displaystyle {v_{f}}_{x}^{2}={v_{i}}_{x}^{2}+2a_{x}\Delta x\,} \)

where

\( {\displaystyle {v_{f}}_{x}} \) is the object's final velocity along the x axis on which the acceleration is constant.
\( {\displaystyle {v_{i}}_{x}} \) is the object's initial velocity along the x axis.
\(a_{x} \) is the object's acceleration along the x axis, which is given as a constant.
\( {\displaystyle \Delta x\,} \) is the object's change in position along the x axis, also called displacement.

This equation is valid along any axis on which the acceleration is constant.

Derivation

Begin with the definition of acceleration:

\({\displaystyle a_{x}={\frac {{v_{f}}_{x}-{v_{i}}_{x}}{\Delta t}}} \)

where \( {\textstyle \Delta t} \) is the time interval. This is true because the acceleration is constant. The left hand side is this constant value of the acceleration and the right hand side is the average acceleration. Since the average of a constant must be equal to the constant value, we have this equality. If the acceleration was not constant, this would not be true.

Now solve for the final velocity:

\({\displaystyle {v_{f}}_{x}={v_{i}}_{x}+a_{x}\Delta t\,\!}

Square both sides to get:

\( {\displaystyle {v_{f}}_{x}^{2}=({v_{i}}_{x}+a_{x}\Delta t)^{2}={v_{i}}_{x}^{2}+2a_{x}{v_{i}}_{x}\Delta t+a_{x}^{2}(\Delta t)^{2}\,\!} (1)

The term \( {\displaystyle (\Delta t)^{2}\,\!} \) also appears in another equation that is valid for motion with constant acceleration: the equation for the final position of an object moving with constant acceleration, and can be isolated:

\( {\displaystyle x_{f}=x_{i}+{v_{i}}_{x}\Delta t+a_{x}{\frac {(\Delta t)^{2}}{2}}} \)

\( {\displaystyle x_{f}-x_{i}-{v_{i}}_{x}\Delta t=a_{x}{\frac {(\Delta t)^{2}}{2}}} \)

\( {\displaystyle (\Delta t)^{2}=2{\frac {x_{f}-x_{i}-{v_{i}}_{x}\Delta t}{a_{x}}}=2{\frac {\Delta x-{v_{i}}_{x}\Delta t}{a_{x}}}} \) (2)

Substituting (2) into the original equation (1) yields:

\( {\displaystyle {v_{f}}_{x}^{2}={v_{i}}_{x}^{2}+2a_{x}{v_{i}}_{x}\Delta t+a_{x}^{2}\left(2{\frac {\Delta x-{v_{i}}_{x}\Delta t}{a_{x}}}\right)} \)

\( {\displaystyle {v_{f}}_{x}^{2}={v_{i}}_{x}^{2}+2a_{x}{v_{i}}_{x}\Delta t+2a_{x}(\Delta x-{v_{i}}_{x}\Delta t)} \)

\( {\displaystyle {v_{f}}_{x}^{2}={v_{i}}_{x}^{2}+2a_{x}{v_{i}}_{x}\Delta t+2a_{x}\Delta x-2a_{x}{v_{i}}_{x}\Delta t\,\!} \)

\( {\displaystyle {v_{f}}_{x}^{2}={v_{i}}_{x}^{2}+2a_{x}\Delta x\,\!} \)