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:
$$ E_{KIN1}(t_1) + E_{KIN2}(t_1) = E_{KIN}(t_2) + E_{V} $$
$$ \frac{m_1}{2}\, v_1^2 + \frac{m_2}{2}\, v_2^2 = \frac{m_1 + m_2}{2}\, v^{‘2} + E_{V} $$
$$ E_V = E_1 – E_2 \qquad
E_V = \frac{m_1 \, m_2}{ 2 \, (m_1 + m_2) } \, (v_1 – v_2)^2 $$
| Work | Work and Energy | Power |