Kinematics in One Dimension

Kinematics & Rates

Close your eyes on a smoothly cruising train and you feel nothing, but the slightest braking registers instantly. That asymmetry is kinematics: describing motion without asking what causes it.

Motion in a straight line uses three core quantities. Position ($x$) is where you are, velocity ($v$) is how position changes, and acceleration ($a$) is how velocity changes:

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

Each layer is the rate of change of the one before it.

Constant Acceleration & Graphs

When acceleration is constant, use the SUVAT equations:

Graphical analysis connects geometry to motion:

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

Free fall is motion under gravity alone, where $a = -g = -9.8\text{ m/s}^2$ (upward positive).

Common pitfall: Negative acceleration does not mean slowing down. A falling ball has $a < 0$ while speeding up. Slowing down means $v$ and $a$ have opposite signs.