Torricelli's equation

Not to be confused with Torricelli's law, Torricelli's theorem, Torricelli's trumpet, or Torricelli point.

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 definition of acceleration:

$a=\frac{v_f-v_i}{t}$

$v_f = v_i + at\,\!$

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 into the original equation yields:

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

$v_f^2 = v_i^2 + 2av_it + 2a(\Delta d - v_it)$

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

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

See also

• Equation of motion

External links

• Torricelli's theorem