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Formula for a Final Velocity for a Constant Acceleration[1]

"Math is dank" - Issac Pascal[2]

There are situations where you are interested in finding the final velocity, but velocity is not a constant number. There is an algebraic equation to find this final velocity without involving calculus.

What you do need to know is:

## Step 1: Time

Consider the following equation:
$V_t = V_o + at$

Using basic algebra, t can be solved as
$t = \frac{V_t-V_o}{a}$

## Step 2: Average Velocity

Also consider what the definition of "average velocity" is: your chance in distance over your change in time
$\bar{V} = \frac{\Delta{x}}{\Delta{t}}$
The change in distance is also known as velocity, making average velocity

A previous equation we learned for x(t) was
$x(t)= x_0+v_0t+\frac{1}{2}at^2$
If we consider that $\Delta{x} = x(t_f) - x(t_i)$ then we can plug those into the above equation and find the following:
$\Delta{x} = V_o(t_f-t_i)+\frac{1}{2}a(t_f^2-t_i^2)$
Do one step of factoring and we get:
$\Delta{x} = V_o(t_f-t_i)+\frac{1}{2}a(t_f+t_i)(t_f-t_i)$
Now consider that $\Delta{t} = t_f-t_i$ and we can go back to the original equation for average velocity.

$\frac{\Delta{x}}{\Delta{t}} = \frac{V_o(t_f-t_i)+\frac{1}{2}a(t_f+t_i)(t_f-t_i)}{t_f-t_i}$
Once you simplify by canceling the bottom with the top, you are left with
$\frac{\Delta{x}}{\Delta{t}} = V_o+\frac{1}{2}a(t_f+t_i)$
Now let us consider just one part of the equation for a moment.
$a(t_f+t_i)$
We can transform this information into something more usable knowing that $V(t) = V_o+at$
Note that because a distributes to two different times, we will do the next operation twice.
$V(t_f) = V_o + at_f$
$V_f - V_o = at_f$
Repeat for $V_i$ and get
$V_i-V_o = at_i$
So, finally,
$a(t_f+t_i) = at_f+at_i = V_f+V_i - 2V_o$
Plugging this back in above we get
$\frac{\Delta{x}}{\Delta{t}} = V_o+\frac{1}{2}( V_f+V_i - 2V_o)$
And as the last step, simply by removing the $V_o$s and end up with
$\frac{\Delta{x}}{\Delta{t}} = \bar{V} = \frac{V_f+V_i}{2}$

## Step 3: Putting it all together

Taking another look at average velocity,
$\bar{V} = \frac{\Delta{x}}{\Delta{t}}$

$\bar{V} = \frac{x_f-x_i}{t_f-t_i}$

Solving for final x we get
$x_f =\bar{V}(t_f-t_i)+x_i$

Now for the fun part! Using the equations for t and for $\bar{V}$ we found in step one and two, we plug those into this equation.

$x_f = x_i + (\frac{V_f+V_i}{2})(\frac{V_f-V_o}{a}-\frac{V_i-V_o}{a})$

$x_f= x_i+(\frac{V_f+V_i}{2})(\frac{V_f-V_i}{a})$

$x_f = x_i + \frac{V_f^2-V_i^2}{2a}$

$x_f-x_i = \frac{V_f^2-V_i^2}{2a}$

$2a(x_f-x_i) = V_f^2-V_i^2$

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

This is not the most fun to derive, but the equation ends up being useful when you have a linear type of velocity.

## Projects

1. Deriving the final velocity equation

P͏̛́͏͎̗͙͡ͅẖ̨̖̩̱̟̣̮̪͎̺͉̲̫͜͜͜ͅy̡҉̶̸̥͚̞͉̗̯͈͟s̷̶̡̥̘̲͇͙̭̝̱͚͙̻͘͠i̷̛͚͕͕̘̩͓̮̲̕͜ͅͅc̵̨̤̼̫͈͚̪̪̞̥̦̯̤̼͉̟̖͚̪̀ͅṣ͇̺͚̼͜͠ ҉̶̥͍̭̤̟i̶͠͏̜̠̟̞̹̠̮͈̺̻̫̭͚̟͟ş̡̝̱̫͔͡͝ ̶̡̯̻͖͓̺̳͘͜a̵̢̧̛͉̩̺͈̤̖̬͖̯͚͠ ̴̵͍̯̱͚͎͇̟̯͡͝͠g̶͘͡͏͉͚̭̫͉̝͟ͅŗ̶̝̲͚̼̼̭̩̗̞̙͠e̶̯̺͓̫͈̜̻̩̕à̶̳̯̦͇̻̤͍͇̹̲t̛̀͘͠͏̱̯̥͓̰̘͕̯̭̟̘̬̼̝͈̩̻̣̲ ̷͞҉҉̤̳̹̳͉̫͉͕̻̹̺̲̰ͅc̷̨͡͏̤͍͖̜l̨͇͍̝̥͓̩̼͇̗̪͉͚̲a̸̕͏̶̶̫̳͔͎͍̦͈s̛̛̝̫͖̗̝̳̭̕͟s͢͟͟҉͓̮̺̭̺͍͍̳͕

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