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

Enter a value for all fields

The **Final Velocity Squared** calculator computes the final velocity a body would achieve after traveling in a straight line some distance at constant acceleration.

**INSTRUCTIONS:** Choose units and enter the following:

- (
**x**) Initial displacement_{i} - (
**x**) Final displacement_{f} - (
**a**) Constant acceleration - (
**V**) Initial velocity_{i}

**Final Velocity Squared (V _{f}^{2}):** The calculator returns the value in Sieverts(sv) where 1 sievert equals one meters squared per seconds squared.

The formula for the final velocity squared is:

V_{f}^{2} = v_{i}^{2}+2?a?(x_{f}-x_{i})

where:

- V
_{f}= final velocity - V
_{i}= initial velocity - a = constant acceleration
- x
_{i}= initial displacement - x
_{f}= final displacement

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