Torricelli's equation

From Wikipedia, the free encyclopedia

 This article does not cite any references or sources. (January 2008)
Please help improve this article by adding citations to reliable sources. Unverifiable material may be challenged and removed.

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 \,

[edit] Derivation

Derivation starts from the equation of the displacement d = d(t) with known initial displacement di and initial velocity vi:

d = d_i + v_i t + \frac {( a t^2 )} {2} \,

We also need the equation for the final velocity vf at time t:

v_f = v_i + a t \,

Isolating time from the second equation, we have:

v_f = v_i + a t \,
v_f - v_i = a t \,
t = \frac {( v_f - v_i )} {a} \,

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

d = d_i + v_i t + \frac {( a t^2 )} {2} \,
d - d_i = v_i \left (\frac {v_f - v_i} {a} \right) + \frac {a} {2} \left(\frac {v_f - v_i} {a} \right)^2 \,
\Delta d = \left (\frac {v_f v_i - v_i^2} {a} \right) + \frac {a} {2} \left(\frac {v_f^2 - 2 v_f v_i + v_i^2} {a^2} \right) \,
\Delta d = \frac {v_f v_i - v_i^2} {a} + \frac {v_f^2 - 2 v_f v_i + v_i^2} {2a} \,
\frac {2a \Delta d} {2a} = \frac {2 v_f v_i - 2 v_i^2} {2a} + \frac {v_f^2 - 2 v_f v_i + v_i^2} {2a} \,
2a \Delta d = 2 v_f v_i - 2 v_i^2 + v_f^2 - 2 v_f v_i + v_i^2 \,
2a \Delta d = - v_i^2 + v_f^2 \,
v_f^2 = v_i^2 + 2 a \Delta d \,

[edit] See also

Equation of motion

[edit] External links