# Final Velocity (from constant a)

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MichaelBartmess.Final Velocity (from constant a)
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This equation computes the square of the final velocity that a body would achieve after traveling in a straight line some distance at constant acceleration. This an illustrative step in calculating the actual v_f based on acceleration and time.  See the derivation below.

The remaining step, to take the square root of both sides of this equation happens in the sister equation:

## INPUTS

• x_i - the initial displacement
• x_f - the final displacement
• a - the constant acceleration
• V_0 - the initial velocity

## DERIVATION

Since acceleration is constant, we know that the final velocity is the sum of the initial velocity and the velocity increase due to the acceleration.  In other words:

[1] V_f = V_i + a *  t

We also know that the distance traveled, d, is the sum of the distance the object would travel at its starting velocity, V_i, plus the distance it would travel while increasing velocity from V_i to V_f:

[2] D = (V_i * t) + (1/2 * (V_f - V_i) * t)

[3] D = t * (V_i + 1/2 * V_f  -  1/2 * V_i)

[4] D = t * 1/2 (V_i + V_f)

[5]    =>   t = (2 * D) / (V_i + V_f)

Substituting [5} into [1]:

[6] V_f   =  V_i + a *  ((2 * D) / (V_i + V_f))

Multiplying both sides by '(V_i + V_f):

[7] V_i *V_f + V_f^2  = V_i^2 + V_i * V_f + 2*A*D

Cancelling term V_i* V_f:

[8] V_f^2 = V_i^2 + 2*a*D,   where D = x_f - x_0

[9] V_f^2 = V_i^2 + 2*a*(x_f - x_0)

This equation [9] computes the resultant V_f^2, which is not useful in most cases, so we want to get the square root of this resultant:

[10] V_f = sqrt(V_i^2 + 2*a*(x_f - x_0))`

Khan Academy's Average velocity for constant acceleration

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