# Torricelli's equation

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:

$v_f^2 = v_i^2 + 2 a \Delta d \,$

where

• $v_f$ is the object's final velocity
• $v_i$ is the object's initial velocity
• $a$ is the object's acceleration
• $\Delta d \,$ is the object's change in position

## Derivation

Begin with the equation for velocity:

$v_f = v_i + at\,\!$

Square both sides to get:

$v_f^2 = (v_i + at)^2 = v_i^2 + 2av_it + a^2t^2\,\!$

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

$d = d_i + v_it + a\frac{t^2}2$
$d - d_i - v_it = a\frac{t^2}2$
$t^2 = 2\frac{d-d_i - v_it}{a} = 2\frac{\Delta d - v_it}{a}$

Substituting this back into our original equation yields:

$v_f^2 = v_i^2 + 2av_it + a^2\left(2\frac{\Delta d - v_it}{a}\right)$
$v_f^2 = v_i^2 + 2av_it + 2a(\Delta d - v_it)$
$v_f^2 = v_i^2 + 2av_it + 2a\Delta d - 2av_it\,\!$
$v_f^2 = v_i^2 + 2a\Delta d\,\!$