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:

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

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

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