Torricelli's equation

From Freepedia

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:

<math> v_f^2 = v_i^2 + 2 a \Delta d \,</math>

This can be derived from these two other formulas:

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

and

<math> v_f = v_i + a t \,</math>

Isolating time from the second equation, we have:

<math> v_f = v_i + a t \,</math>

<math> v_f - v_i = a t \,</math>

<math> t = \frac {( v_f - v_i )} {a} \,</math>

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

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

<math> 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 \,</math>

<math> \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) \,</math>

<math> \Delta d = \frac {v_f v_i - v_i^2} {a} + \frac {v_f^2 - 2 v_f v_i + v_i^2} {2a} \,</math>

<math> \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} \,</math>

<math> 2a \Delta d = 2 v_f v_i - 2 v_i^2 + v_f^2 - 2 v_f v_i + v_i^2 \,</math>

<math> 2a \Delta d = - v_i^2 + v_f^2 \,</math>

<math> v_f^2 = v_i^2 + 2 a \Delta d \,</math>

See also

Equation of motion