An object's position, velocity, and acceleration can be modeled as functions of time, $x(t),$ $v(t),$ and $a(t).$ These functions are related through **differentiation**. Velocity is the derivative of position with respect to time, and acceleration is the derivative of velocity with respect to time:
$\begin{align}v(t)=\dfrac{dx}{dt} \\\\ a(t)=\dfrac{dv}{dt}\end{align}$
The kinematic quantities are also related through **integration**. An object's change in position (displacement) during an interval is equal to all the small displacements which accumulate during the interval, which is represented by the signed area bounded on a graph of $v(t).$ Adding this area to the object's initial position gives its position at time $t.$ Similarly, finding the area bounded on a graph of $a(t)$ and adding it to the object's initial velocity gives its velocity at time $t:$
$\begin{align}x(t)=x(0)+\displaystyle{\int_{0}^{t}v(\tau)\,d\tau} \\\\ v(t)=v(0)+\displaystyle{\int_{0}^{t}a(\tau)\,d\tau}\end{align}$
[[☃ explanation 1]]
So, we can relate the kinematic quantities through rates of change or through accumulated change. We can also use these relationships to derive additional equations relating $x,$ $v,$ and $a$ which can be useful in certain situations. Let's derive these equations for a an object moving with constant acceleration.
##Modeling constant acceleration
Consider an object moving in one dimension with constant acceleration. A coordinate system for the dimension of motion with an origin is shown below.
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The graph of $a(t)$ for an object moving with a constant acceleration of $a_0$ is shown below.
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##From $a(t)$ to $v(t)$ to $x(t)$
Now that we have a general equation for $a(t)$ for an object moving with constant acceleration, let's use it to derive an equation for $v(t).$
[[☃ graded-group-set 1]]
The form of $v(t)=\blueE{v_0}+\redE{a_0}t$ matches the general $y(x)=\blueE{b}+\redE{m}x.$ So, we can predict that the graph of $v(t)$ for an object moving with constant acceleration is a straight line. The constant $\blueE{v_0}$ is the $y$-intercept of the line, representing the velocity at $t=0.$ And $\redE{a_0}$ is the slope of the line, representing the rate of change of the velocity. The graph of $v(t)$ for an object moving with constant acceleration and an initial velocity $v_0$ is shown below.
[[☃ image 2]]
Now that we have a general equation for $v(t)$ for an object moving with constant acceleration, let's use it to derive an equation for $x(t).$
[[☃ graded-group-set 2]]
The form of $x(t)=\goldE{x_0}+\blueE{v_0}t+\redE{\tfrac{1}{2}a_0}t^2$ matches the general $y(x)=\goldE{c}+\blueE{b}x+\redE{a}x^2.$ So, we can predict that the graph of $x(t)$ for an object moving with constant acceleration is a parabola. The constant $\goldE{x_0}$ is the $y$-intercept of the parabola, representing the position at $t=0.$ The graph of $x(t)$ for an object moving with constant acceleration and an initial position of $x_0$ is shown below.
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##Eliminating $t$
The equations we derived above are functions of time. They include the variable $t,$ so they're useful if time is one of the variables we explicitly want to consider.
$\begin{align}&v(t)=v_0 + a_0t \\ &x(t)=x_0 + v_0t + \tfrac{1}{2}a_0t^2\end{align}$
There are also cases when it's helpful to relate position, velocity, and acceleration for an object moving with constant acceleration, *without* explicitly including $t$. We can use the two equations we derived to derive a third equation which does not include $t.$
[[☃ graded-group-set 1]]
##Summary
We began this article with fundamental definitions relating an object's position, velocity, and acceleration over time. These definitions hold true for *any* motion in a single dimension.
$\begin{align}&v(t)=\dfrac{dx}{dt} \\ &a(t)=\dfrac{dv}{dt} \\ &x(t)=x(0)+\displaystyle{\int_{0}^{t}v(\tau)\,d\tau} \\ &v(t)=v(0)+\displaystyle{\int_{0}^{t}a(\tau)\,d\tau}\end{align}$
Then, we considered the specific case of an object moving with constant acceleration, which we modeled as $a(t)=a_0.$ We used this model along with the fundamental definitions above to derive three **kinematic equations** which relate $x,$ $v,$ and $a_0$ in a single dimension. Since we began deriving these equations using the model $a(t)=a_0,$ they are *only valid for motion with constant (uniform) acceleration.*
$\begin{align}&v=v_0 + a_0t \\ &x=x_0 + v_0t + \tfrac{1}{2}a_0t^2 \\ &v^2=v_0^2+2a_0(x-x_0)\end{align}$
We can represent these same equations with a few notational changes. We can replace $a_0$ with $a,$ keeping in mind that $a$ is a constant in these equations. We can also add subscript $x\text{'s}$ to the kinematic variables to remind us that the equations apply to motion in a single dimension.
$\boxed{\begin{align}&v_x=v_{x0} + a_xt \\ &x=x_0 + v_{x0}t + \tfrac{1}{2}a_xt^2 \\ &v_x^2=v_{x0}^2+2a_x(x-x_0)\end{align}}$
[[☃ explanation 1]]