Velocity-Time Graphs and Graphical Analysis

A velocity-time (v-t) graph plots the instantaneous velocity of a body along the vertical axis against time along the horizontal axis. It is, in a definite sense, more information-rich than a position-time graph for our present purposes, because it lets us read off both acceleration (from its slope) and displacement (from the area beneath it) directly.

Slope of a v-t graph gives acceleration. Since instantaneous acceleration is defined as $a = dv/dt$ (Section 2.5), and the derivative at a point is the slope of the tangent to the graph at that point, the slope of a v-t graph at any instant is exactly the instantaneous acceleration at that instant. For uniformly accelerated motion, $a$ is constant, so the v-t graph is a straight line whose (constant) slope is that constant acceleration — steeper lines mean larger acceleration, and a line sloping downward corresponds to a negative (deceleration/retarding) acceleration.

Area under a v-t graph gives displacement. This is the more subtle — and more powerful — reading. Because $v = dx/dt$, we have $dx = v\,dt$, and integrating both sides between times $t_1$ and $t_2$,

The integral on the right-hand side is, by the fundamental geometric meaning of a definite integral, exactly the area enclosed between the v-t curve and the time axis over the interval $[t_1, t_2]$ (counted as negative area for any stretch where $v$ is negative). So:

Applying this to uniformly accelerated motion. For a body with constant acceleration $a$, starting with velocity $u$ at $t = 0$ and reaching velocity $v$ at time $T$, the v-t graph is a straight line rising (or falling) from $u$ to $v$. The region under this line, above the time axis, between $t=0$ and $t=T$, is a trapezium with parallel sides of length $u$ and $v$ and width $T$. Its area is

which is exactly the displacement covered in time $T$. This graphical route reproduces the average-of-initial-and-final-velocity kinematic relation used later in Section 2.8, entirely from the shape of the graph, with no separate calculus required. The accompanying figure shows exactly this trapezoidal shaded region. …