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
$$ \vec{s}(t) = x(t) \, \vec{e}_x + y(t) \, \vec{e}_y + z(t) \, \vec{e}_z $$
$$ \vec{v}(t) = v_x(t) \, \vec{e}_x + v_y(t) \, \vec{e}_y + v_z(t) \, \vec{e}_z $$
$$ \vec{a}(t) = a_x(t) \, \vec{e}_x + a_y(t) \, \vec{e}_y + a_z(t) \, \vec{e}_z $$
| Kinematics | Rotational Motion |