Torricelli's equation

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:

:$v\_f^2\; =\; v\_i^2\; +\; 2\; a\; Delta\; d\; ,$

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:

$d\; =\; d\_i\; +\; v\_it\; +\; afrac\{t^2\}2$

$d\; -\; d\_i\; -\; v\_it\; =\; afrac\{t^2\}2$

$t^2\; =\; 2frac\{d-d\_i\; -\; v\_it\}\{a\}\; =\; 2frac\{Delta\; d\; -\; v\_it\}\{a\}$

Substituting this back into our original equation yields:

$v\_f^2\; =\; v\_i^2\; +\; 2av\_it\; +\; a^2left(2frac\{Delta\; d\; -\; v\_it\}\{a\}\; ight)$

$v\_f^2\; =\; v\_i^2\; +\; 2av\_it\; +\; 2a(Delta\; d\; -\; v\_it)$

$v\_f^2\; =\; v\_i^2\; +\; 2av\_it\; +\; 2aDelta\; d\; -\; 2av\_it,!$

$v\_f^2\; =\; v\_i^2\; +\; 2aDelta\; d,!$

ee also

Equation of motion

External links

* [http://demonstrations.wolfram.com/TorricellisTheorem/ Torricelli's Theorem] by Ernest Lee, The Wolfram Demonstrations Project.