# Formulas – Energy

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Formula for relativistic kinetic energy
The formulas for kinetic energy are based on Einstein’s relationship for energy:
Total energy:

$E=mc^2$

and

Rest energy (energy at rest):

$E_0=m_0c^2$

Where:

$m = m_0\gamma$

m = relativistic mass
m0 = rest mass
c = speed of light in vacuum

$\gamma =\dfrac{1}{\sqrt{1-(v/c)^2}}$

Total energy may also be expressed as:

$E=E_0 + E_k$

Relativistic kinetic energy may therefore be calculated from:

$E_k=E - E_0$

and

$E_k=mc^2 - m_0c^2$

$E_k=m_0 \gamma c^2 - m_0c^2$

$E_k=\dfrac{m_0c^2}{\sqrt{1-(v/c)^2}} - m_0c^2$

At very low speeds, compared to the speed of light, the classical formula for kinetic energy (below) aligns well with the formula for relativistic kinetic energy, something we can see by applying the binomial theorem to the formula for relativistic kinetic energy (above).

$E_k=\dfrac{m_0c^2}{\sqrt{1-(v/c)^2}} - m_0c^2$

$E_k \approx \dfrac{1}{2}m_0c^2 + \dfrac{3}{8 } \dfrac{m_0v^4}{c^2 } + \dfrac{5}{16 } \dfrac{m_0v^6}{c^4 } + \cdots - m_0c^2 = \dfrac{1}{2}m_0v^2$

Classical formula for kinetic energy
Ek = Et+ Er
Et = Translational kinetic energy
Er = Rotational kinetic energy

Formula for translational kinetic energy

$E_t=\dfrac{1}{2}mv^2$

m = mass (Kg)
v = velocity (m/s metres per second)
Ek = Resulting energy is measured in joules

Formula for rotational kinetic energy

$E_r=\dfrac{1}{2}Iw^2$

I = Moment of inertia (around the axis of rotation)
ω = Angular velocity = 2πf
f = Revolutions/sec

Formula for gravitational potential energy.

$E_{pg}= mgh$
m = Mass
g = Gravitational acceleration (9,8 m/s2 close to earth)
h = Height above the reference point

Formula for elastic potential energy for a linear spring.

$E_{pe}=\dfrac{1}{2}kx^2$
k = Spring constant
x = Amount of stretch or compression

Formula for thermal energy

$Q = \int_{t_1}^{t_2}mc\Delta t = m\int_{t_1}^{t_2}c\Delta t$

Q = Thermal energy of a substance or a system
m = The mass of the substance or system
c= The specific heat capacity of the substance or system.
T = the absolute temperature of the substance or system
Δt = Temperature difference

For practical purposes the average specific heat capacity (cm) may be used, the formula then is:
Q = mcm (t2 – t1)

Formula electrical energy
Energy (Joule) = Power (Watt) x Time (Second)
Power (Watt) = Energy(Joule) / Time(Second)
1 Watt = 1 Joule / Second.

Electrical energy may be defined by the work (W) carried out or needed to move electrically charged particles.

W = UIt (Joule)
U = Differential potential (Volt)
I = current (Ampere) (Columb per second)
R = Resistanse (Ohms Ω)
t = time (second):

Power
P (Power) = W/t = UIt/t = Ui (volts x ampere) (Watt)
P = R x I2
P = U2/R

Current
I = P/U (ampere)
I = U/R
I = (P/R)1/2

Electrical potential
U = RI
U = P/I
U = (PR)1/

Resistance
R = U/I (ohm`s law)
R= P/I
R = P/I2

Energy of an electric field
The work done in establishing the electric field, and hence the amount of energy stored, is:

$W= \dfrac{1}{2}CV^2 + \dfrac{1}{2}VQ$

Q = Charge stored
V = Voltage across the capacitor
C = Capacitance

Formula for electromagnetic energy
The energy for one individual photon is:

$E = hv = \dfrac{hc}{\lambda} (Joule)$

or if angular frequency is used:

$E = \hbar w$

ω  = 2πv
ν   = Frequency (cycles/second)
λ   = Wavelength (metres)
c   = Speed of light (metres/second)

1 Hz = 1 hertz; cycle per second (frequency)
1 nm = 10-9 m, nanometre (for wavelength of IR, visible, UV and X-rays).
1 pm = 10-12 m, picometer (for X-rays and gamma rays).

To calculate the energy giving the result in everyday quantities we need to calculate the combined energy for larger number of particles.

Formula for sound energy
The total sound energy will equal the maximum kinetic energy:

$E= \dfrac{1}{2}mv^2 = \dfrac{1}{2}m(A\omega)^2$

= density of the medium the sound waves travel through
Aω = the maximum transverse speed of particles

$A = \dfrac{\nu}{2\pi}$

A = amplitude

Formula for nuclear energy

Mass defect and nuclear binding energy

$E= mc^2$ or

$m= \dfrac{c^2}{E}$

We first need to calculate the mass defect to be able to calcutlate the potential for releasing energy when fission takes place.

Mass defect
Mass c (combined mass)    = MP + MN (Mass Neutron)
MP =Mass Proton = nP*amuP
MN =Mass Neutron = nN*amuN
Dm = Mass c – MassBM

Mass defect into kg
Dm(amu) * 1.6606 x 10-27 kg/nucleus
1amu = 1.6606 x 10-27 kg

Mass defect into energy
c = 2.9979 x 108 m/s
E = mc2 = (Dm(amu) *1.6606* 10-27 kg/nucleus) * (2.9979 x 108 m/s)2
E = DM*1,4924483 *10-10 J/nucleus

E= DM*1,4924483 *10-10 J/nucleus * 6.022 x 1023 nuclei/mol* (1 kJ/1000 J) * = DM*8,9875 1010 kJ/mol of nuclei.

Avogadro’s Number = 6.022 x 1023 nuclei/mol