Torricelli's equation

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Torricelli's equation is an equation 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 \Delta d \,

This can be derived from these two other formulas:

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

and

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

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