*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

m_{0} = 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

(a+b)^n=a ^n+na^{n-1}b +\dfrac{n(n-1)}{2}a^{n-2} b^2+\cdots+b^n

For any value of n, the value of the n^{th} power of a binomial is given by the above equation

*Classical formula for kinetic energy*Ek = E

_{t}+ E

_{r}

E

_{t}= Translational kinetic energy

E

_{r}= 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)

E

_{k}= 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

*E_{pg}= mgh*

**Formula for gravitational potential energy.**

*m*= Mass

*g*= Gravitational acceleration (9,8 m/s2 close to earth)

*h*= Height above the reference point

*E_{pe}=\dfrac{1}{2}kx^2*

**Formula for elastic potential energy for a linear spring.**

*k*= Spring constant

*x*= Amount of stretch or compression

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

**Formula for thermal energy**

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)

**Energy (Joule) = Power (Watt) x Time (Second)**

*Formula electrical energy*

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

*m*= 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