Translational Motion

$s$ $\mathrm{m}$ distance, way travelled
$v$ $\mathrm{m/s}$ velocity
$a$ $\mathrm{m/s^2}$ acceleration
$t$ $\mathrm{s}$ time
$s_0$ $\mathrm{m}$ initial position
$v_0$ $\mathrm{m/s}$ initial velocity
$a_0$ $\mathrm{m/s^2}$ initial acceleration
$e$ unit vector

Uniform Movement

Conditions: $a = 0$ $v \neq 0 = \mathrm{const.}$

$$ v = \int\! a \;\mathrm{d}t = v_0 = \mathrm{const.} $$

$$ v = \frac{ \mathrm{d}s }{ \mathrm{d}t } $$

$$ s = \int\! v \;\mathrm{d}t = v_0 \, t + s_0 $$

Uniformly accelerated Motion

Conditions: $a = a_0 > 0 = \mathrm{const.}$

$$ v = \int\! a_0 \;\mathrm{d}t = a_0 \, t + v_0 $$

$$ s = \int\! v \;\mathrm{d}t = \int\! ( a_0 \, t + v_0) \;\mathrm{d}t
= \frac{a_0}{2} \, t^2 + v_0 \, t + s_0 $$

Irregular accelerated Motion

Conditions: $a = a(t) \neq \mathrm{const.}$

$$ v = \int\! a(t) \;\mathrm{d}t = a(t) \, t + v_0 $$

$$ s = \int\! v(t) \;\mathrm{d}t = \int\! ( a(t) \, t + v_0) \;\mathrm{d}t
= \frac{a(t)}{2} \, t^2 + v_0 \, t + s_0 $$

Translational Motion in general

$$ s = \int\limits_{t_1}^{t_2}\! v(t) \;\mathrm{d}t $$

$$ v = \int\limits_{t_1}^{t_2}\! a(t) \;\mathrm{d}t \qquad
v = \frac{ \mathrm{d}s }{ \mathrm{d}t } = s'(t) = \dot{s} $$

$$ a = \frac{ \mathrm{d}v }{ \mathrm{d}t } = v'(t) = \dot{v} \qquad
a = \frac{ \mathrm{d^2}s }{ \mathrm{d}t^2 } = \ddot{s} $$

Acceleration with $v_0 = 0$

$$ s = \frac{a}{2} \, t^2 \qquad
s = \frac{v}{2} \, t $$

$$ v = a \, t \qquad
v = \sqrt{ 2 \, a \, s } $$

Acceleration with $v_0 \neq 0$

$$ s = \frac{a}{2} \, t^2 + v_0 \, t \qquad
s = \frac{v + v_0}{2} \, t $$

$$ v = a \, t + v_0 \qquad
v = \sqrt{ 2 \, a \, s + v_0^2} $$

Average velocity
$$ \overline{v} = \frac{\Delta s}{\Delta t} $$

Average acceleration
$$ \overline{a} = \frac{\Delta v}{\Delta t} $$

2D-Translational Motion

To calculate separately

$$ s(t) = \begin{array}{c} x(t) \\ y(t) \end{array} = \left( \begin{array}{c} x \\ y \end{array}\right) $$

$$ x(t) = v_x(t) \, t + x_0 \qquad
y(t) = v_y(t) \, t + y_0 $$

3D-Translational Motion

$$ \vec{s}(t) = x(t) \, \vec{e}_x + y(t) \, \vec{e}_y + z(t) \, \vec{e}_z $$

$$ \vec{v}(t) = v_x(t) \, \vec{e}_x + v_y(t) \, \vec{e}_y + v_z(t) \, \vec{e}_z $$

$$ \vec{a}(t) = a_x(t) \, \vec{e}_x + a_y(t) \, \vec{e}_y + a_z(t) \, \vec{e}_z $$