This equation was created by Evangelista Torricelli to find the final velocity of a moving object without having a known time interval and was named after him.

The equation itself is:

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

This can be derived from these two other formulas:

$d\; =\; d\_i\; +\; v\_i\; t\; +\; \backslash frac\; \{(\; a\; t^2\; )\}\; \{2\}\; \backslash ,$

and

$v\_f\; =\; v\_i\; +\; a\; t\; \backslash ,$

Isolating time from the second equation, we have:

$v\_f\; =\; v\_i\; +\; a\; t\; \backslash ,$

$v\_f\; -\; v\_i\; =\; a\; t\; \backslash ,$

$t\; =\; \backslash frac\; \{(\; v\_f\; -\; v\_i\; )\}\; \{a\}\; \backslash ,$

And now, substituting this equation into the first one, we have:

$d\; =\; d\_i\; +\; v\_i\; t\; +\; \backslash frac\; \{(\; a\; t^2\; )\}\; \{2\}\; \backslash ,$

$d\; -\; d\_i\; =\; v\_i\; \backslash left\; (\backslash frac\; \{v\_f\; -\; v\_i\}\; \{a\}\; \backslash right)\; +\; \backslash frac\; \{a\}\; \{2\}\; \backslash left(\backslash frac\; \{v\_f\; -\; v\_i\}\; \{a\}\; \backslash right)^2\; \backslash ,$

$\backslash Delta\; d\; =\; \backslash left\; (\backslash frac\; \{v\_f\; v\_i\; -\; v\_i^2\}\; \{a\}\; \backslash right)\; +\; \backslash frac\; \{a\}\; \{2\}\; \backslash left(\backslash frac\; \{v\_f^2\; -\; 2\; v\_f\; v\_i\; +\; v\_i^2\}\; \{a^2\}\; \backslash right)\; \backslash ,$

$\backslash Delta\; d\; =\; \backslash frac\; \{v\_f\; v\_i\; -\; v\_i^2\}\; \{a\}\; +\; \backslash frac\; \{v\_f^2\; -\; 2\; v\_f\; v\_i\; +\; v\_i^2\}\; \{2a\}\; \backslash ,$

$\backslash frac\; \{2a\; \backslash Delta\; d\}\; \{2a\}\; =\; \backslash frac\; \{2\; v\_f\; v\_i\; -\; 2\; v\_i^2\}\; \{2a\}\; +\; \backslash frac\; \{v\_f^2\; -\; 2\; v\_f\; v\_i\; +\; v\_i^2\}\; \{2a\}\; \backslash ,$

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

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

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

See also