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`V_f^2 = V_i ^2+2* a *( x_f - x_i )`

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This equation computes the final velocity a body would achieve after traveling in a straight line some distance at constant acceleration.

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

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`

So, finally:

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

Khan Academy's Average velocity for constant acceleration