Rectangular (Cartesian) Motion

Position

$$x(t) = \int v(t)\,dt$$

Velocity

$$v(t) = x'(t) = \int a(t)\,dt$$

Acceleration

$$a(t) = v'(t) = x''(t)$$

Speed

$|v(t)|$, always non-negative

Displacement

$\int_a^b v(t)\,dt$, net change in position (can be negative)

Total Distance

$\int_a^b |v(t)|\,dt$, always non-negative

New Position

$$s(a) + \int_a^b v(t)\,dt$$

Average Velocity

$$\frac{x(t_2)-x(t_1)}{t_2-t_1}$$

Speed increasing

when $a(t)$ and $v(t)$ have the same sign

Speed decreasing

when $a(t)$ and $v(t)$ have opposite signs

Particle at rest

$$v(t) = 0$$

Particle moves right/up

v(t) > 0

Particle moves left/down

v(t) < 0