Torricelli's equation

Not to be confused with Torricelli's Law or Torricelli's Theorem.

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:

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

Derivation

Begin with the equation for velocity:

$v_{f}=v_{i}+at\,\!$

Square both sides to get:

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

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

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

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

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

Substituting this back into our original equation yields:

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

$v_{f}^{2}=v_{i}^{2}+2av_{i}t+2a(\Delta d-v_{i}t)$

$v_{f}^{2}=v_{i}^{2}+2av_{i}t+2a\Delta d-2av_{i}t\,\!$

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

See also

• Equation of motion

External links

• Torricelli's Theorem by Ernest Lee, The Wolfram Demonstrations Project.