# Torricelli's equation

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In physics, Torricelli's equation is an equation created by Evangelista Torricelli to find the final velocity of an object moving with a constant acceleration without having a known time interval.

The equation itself is:

${\displaystyle v_{f}^{2}=v_{i}^{2}+2a\Delta d\,}$

where

• ${\displaystyle v_{f}}$ is the object's final velocity
• ${\displaystyle v_{i}}$ is the object's initial velocity
• ${\displaystyle a}$ is the object's acceleration
• ${\displaystyle \Delta d\,}$ is the object's change in position

## Derivation

Begin with the definition of acceleration:

${\displaystyle a={\frac {v_{f}-v_{i}}{t}}}$
${\displaystyle v_{f}=v_{i}+at\,\!}$

Square both sides to get:

${\displaystyle v_{f}^{2}=(v_{i}+at)^{2}=v_{i}^{2}+2av_{i}t+a^{2}t^{2}\,\!}$

The term ${\displaystyle t^{2}\,\!}$ appears in the equation for displacement, and can be isolated:

${\displaystyle d=d_{i}+v_{i}t+a{\frac {t^{2}}{2}}}$
${\displaystyle d-d_{i}-v_{i}t=a{\frac {t^{2}}{2}}}$
${\displaystyle t^{2}=2{\frac {d-d_{i}-v_{i}t}{a}}=2{\frac {\Delta d-v_{i}t}{a}}}$

Substituting this into the original equation yields:

${\displaystyle v_{f}^{2}=v_{i}^{2}+2av_{i}t+a^{2}\left(2{\frac {\Delta d-v_{i}t}{a}}\right)}$
${\displaystyle v_{f}^{2}=v_{i}^{2}+2av_{i}t+2a(\Delta d-v_{i}t)}$
${\displaystyle v_{f}^{2}=v_{i}^{2}+2av_{i}t+2a\Delta d-2av_{i}t\,\!}$
${\displaystyle v_{f}^{2}=v_{i}^{2}+2a\Delta d\,\!}$