Kinematics in One Dimension

Close your eyes in a smoothly cruising train and you cannot tell how fast you are going — but the slightest braking, you feel instantly. That asymmetry is the heart of kinematics: position is where you are, velocity is how position changes, acceleration is how velocity changes. Each layer is the rate of change of the one above.

Kinematics describes motion without asking what causes it. For motion along a straight line, three quantities matter: position $x$, velocity $v$, and acceleration $a$.

Definitions

$$v = \frac{dx}{dt}, \qquad a = \frac{dv}{dt} = \frac{d^2x}{dt^2}$$

Constant-acceleration equations (the "SUVAT" equations):

Free fall is the special case where $a = -g = -9.8\;\text{m/s}^{2}$ (taking upward as positive).

Graphical analysis

• Slope of $x$-$t$ graph = velocity.
• Slope of $v$-$t$ graph = acceleration.
• Area under $v$-$t$ graph = displacement.

Tip: Choose a sign convention (e.g. upward positive) at the start and stick with it throughout the problem.

Common pitfall: "Negative acceleration" does not automatically mean slowing down. A ball falling faster and faster has $a < 0$ (downward) while its speed grows. Slowing down means $v$ and $a$ point in opposite directions — check signs, not words.