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Here we will take a look at the derivation of the following kinematics equation:

This equation in its above form is not solved for any particular variable.

Notice that it relates four quantities: ** final
velocity**, v

What is important to notice is that the
quantity of time is not present in this equation.
We say, therefore, that the equation is ** time
independent**.

This equation is often useful in kinematics
problems where you do not know the ** time period** of the

We derive this equation by combining the other two kinematics equations in this section, and, through substitutions, eliminating the time variable.

Here are the other two equations; first,
the ** displacement equation**:

And second, the ** velocity equation**:

For the substitution this ** velocity
equation** is rearranged so that it is

The above expression for t can be ** substituted
into the displacement equation**, and the resultant
equation can be simplified and arranged till our time
independent equation takes form.

The algebra for this is a bit more complicated than the algebra in our other examples. We will step through it as follows.

Let's start here:

In the above equation we will substitute the following expression for t:

This gives us:

And on the right side of the equation in the second term, since the square of a fraction equals the square of the numerator over the square of the denominator, we get:

Multiplying each side by a,

Carefully canceling where a/a=1 and a^{2}/a^{2}=1,

We get:

Now, on the right side, first term we
distribute v_{o} into the parentheses and get:

On the right side, third term, by expanding
(v_{f} - v_{o})^{2} we get:

For reasons that are really only cosmetic,
inside of that expansion we will note that v_{f}v_{o}=v_{o}v_{f}:

On the right side, third term we will distribute the 1/2 into the parentheses:

On the right side noting that v_{o}v_{f}
cancels with -v_{o}v_{f} we get:

Now, multiplying each side by 2:

On the right side, since -2v_{o}^{2}+v_{o}^{2}=-v_{o}^{2}:

Commuting terms on the right side we get:

And, lastly, we rearrange by switching the left and the right sides of the equation:

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