Position-Time Graphs

A position-time (x-t) graph plots the position $x$ of a body along the vertical axis against time $t$ along the horizontal axis. Because instantaneous velocity is defined as $v = dx/dt$ (Section 2.4), and the derivative of a function at a point is geometrically the slope of the tangent to its graph at that point, an x-t graph lets us read off velocity directly from its shape, without doing any separate calculation.

Case 1 — a body at rest. If a body does not move, its position $x$ stays the same for all $t$. The x-t graph is a horizontal straight line. Its slope is zero everywhere, correctly reflecting that $v = 0$ throughout.

Case 2 — a body moving with uniform velocity. If a body moves with a constant velocity $v$, its position increases (or decreases) linearly with time: $x(t) = x_0 + vt$. The x-t graph is a straight line with a constant slope equal to $v$. A steeper line corresponds to a larger speed; a line sloping downward to the right corresponds to a negative velocity (motion in the negative direction). This is the case illustrated in the accompanying figure, where the constant slope of the line is explicitly identified as the (constant) velocity.

Case 3 — a body undergoing uniformly accelerated motion. If a body's velocity itself increases (or decreases) at a constant rate — uniformly accelerated motion, taken up formally in Section 2.8 — then $x(t) = x_0 + ut + \tfrac{1}{2}at^2$ is a quadratic function of time, and its graph is a parabola. If the acceleration is positive, the parabola curves upward (is concave up) and gets progressively steeper — the tangent drawn at a later time has a larger slope than the tangent drawn at an earlier time, correctly showing that the instantaneous velocity is increasing with time. This is the case shown in the second accompanying figure, where two tangents drawn at different times visibly differ in steepness.

Reading velocity from an x-t graph in general. For any x-t graph, however irregular, the instantaneous velocity at a given instant is the slope of the tangent to the curve drawn at that instant, while the average velocity over an interval $[t_1, t_2]$ is the slope of the straight chord joining the two points $(t_1, x_1)$ and $(t_2, x_2)$ on the graph — this is exactly the graphical picture underlying the average-velocity definition of Section 2.3, now expressed geometrically. …