## Energy

 $E$ $\mathrm{J}$ energy $E_V$ $\mathrm{J}$ energy loss $m$ $\mathrm{kg}$ mass $s$ $\mathrm{m}$ distance, way travelled $h$ $\mathrm{m}$ height $v$ $\mathrm{m/s}$ velocity $k$ $\mathrm{N/m}$ spring constant $J$ $\mathrm{kg\,m^2}$ moment of inertia $\omega$ $\mathrm{s^{-1}}$ angular velocity

### Potential Energy

$$E_{POT} = m \int\limits_{h_1}^{h_2} \! g \;\mathrm{d}h$$

 Condition: $g = \mathrm{const.}$

$$E_{POT} = m \, g \, h$$

### Kinetic Energy

$$E_{KIN} = \frac{m}{2} \, v^2 \qquad E_{KIN} = \frac{m}{2} \, \left( v_1^2 – v_0^2 \right)$$

### Tensional Energy

Potential energy:

$$E_{POT} = \frac{1}{2} \, k \, s^2$$

### Rotational Energy

$$E_{ROT} = \frac{1}{2} \, J \, \omega^2$$

Rolling wheel:
$$E_{KIN} = E_{ROT} + E_{TRANS}$$

### Principle of Conservation of Energy

$$E = \sum_{i=1}^n E_i = \mathrm{const.}$$

Condition: conservative forces (lossless):
$$\sum E_{POT} + \sum E_{KIN} = \mathrm{const.}$$

Condition: dissipative forces (loss due to friction):
$$\sum E_{POT} |_{t_0} + \sum E_{KIN} |_{t_0} = \sum E_{POT} |_{t_1} + \sum E_{KIN} |_{t_1} + E_V$$

### Mechanical Impacts

Elastic impact:

$$E_{KIN1}(t_1) + E_{KIN2}(t_1) = E_{KIN1}(t_2) + E_{KIN2}(t_2)$$

$$\frac{m_1}{2}\, v_1^2 + \frac{m_2}{2}\, v_2^2 = \frac{m_1}{2}\, v_1^{‘2} + \frac{m_2}{2}\, v_2^{‘2}$$

Inelastic collision:

$$E_{KIN1}(t_1) + E_{KIN2}(t_1) = E_{KIN}(t_2) + E_{V}$$

$$\frac{m_1}{2}\, v_1^2 + \frac{m_2}{2}\, v_2^2 = \frac{m_1 + m_2}{2}\, v^{‘2} + E_{V}$$

$$E_V = E_1 – E_2 \qquad E_V = \frac{m_1 \, m_2}{ 2 \, (m_1 + m_2) } \, (v_1 – v_2)^2$$