Representing Motion

This topic covers constant-acceleration motion in one dimension. Three kinematic equations handle almost every traditional problem: $v_x = v_{x0} + a_x t$, $x = x_0 + v_{x0}t + \tfrac{1}{2}a_x t^2$, and $v_x^2 = v_{x0}^2 + 2a_x(x - x_0)$. Free fall near Earth uses $g \approx 10$ m/s$^2$, where $g$ is a positive magnitude: with upward taken as positive the acceleration component is $a_y = -g$. Motion graphs show the same physics: the slope of $x(t)$ is velocity, the area under $v(t)$ is displacement. When $a$ varies in time, the integral form takes over: $\Delta x = \int v_x\,dt$ and $\Delta v_x = \int a_x\,dt$.

One mistake sits underneath most of the others: reading a motion graph as a picture of the trajectory, so a downward-opening $x$-vs-$t$ parabola becomes a ball tossed into the air. Everything the diagnostic checks follows from that slip — distance versus displacement on a reversing trip, position versus displacement on a graph, the graph-as-picture reading itself, $v = 0$ at an apex misread as $a = 0$, and the sign that rides along with a reversed integration bound.