When a particle has erratic or changing motion then its position, velocity, and acceleration cannot be described by a single continuous mathematical function along the entire path. Instead, a series of functions will be required to specify the motion at different intervals. For this reason, it is convenient to represent the motion as a graph.
If a graph of the motion that relates any two of the variables $s$, $v$, $a$, $t$ can be drawn, then this graph can be used to construct subsequent graphs relating two other variables since the variables are related by the differential relationships $v = ds/dt$, $a = dv/dt$, or $a ds = v dv$. Several situations occur frequently.
To construct the $v–t$ graph given the $s–t$ graph, the equation $v = ds/dt$ should be used, since it relates the variables $s$ and $t$ to $v$. This equation states that
$$ \frac{ds}{dt} = v $$
slope of $s–t$ graph $=$ velocity
For example, by measuring the slope on the $s–t$ graph when $t = t_1$, the velocity is $v_1$. The $v–t$ graph can be constructed by plotting this and other values at each instant.
If the $s–t$ curve for each interval of motion can be expressed by a mathematical function $s = s(t)$, then the equation of the $v–t$ graph for the same interval can be obtained by differentiating this function with respect to time since $v = ds/dt$.
The $a–t$ graph can be constructed from the $v–t$ graph in a similar manner, since
$$ \frac{dv}{dt} = a $$
slope of $v–t$ graph $=$ acceleration
The equation of the $a–t$ graph for the same interval can be determined by differentiating $v = v(t)$ since $a = dv/dt$.
Since differentiation reduces a polynomial of degree $n$ to that of degree $n − 1$, then if the $s–t$ graph is parabolic (a second-degree curve), the $v–t$ graph will be a sloping line (a first-degree curve), and the $a–t$ graph will be a constant or a horizontal line (a zero-degree curve).
If the $a–t$ graph is given, the $v–t$ graph may be constructed using $a = dv/dt$, written as
$$ \Delta v = \int a \, dt $$
change in velocity $=$ area under $a–t$ graph
Hence, to construct the $v–t$ graph, we begin with the particle's initial velocity $v_0$ and then add to this small increments of area $(Δv)$ determined from the $a–t$ graph. In this manner successive points, $v_1 = v_0 + Δ_v$, etc., for the $v–t$ graph are determined.