## 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

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

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

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