1. Introduction

In this section, we will cover the foundational equations related to kinetic energy, mass-energy equivalence, and photon energy.

1.1. Kinetic Energy and Mass

The kinetic energy (\(K\)) of an object in motion is given by:

Where:

• \(K\) is the kinetic energy (in joules, J),
• \(m\) is the mass (in kilograms, kg),
• \(v\) is the velocity of the object (in meters per second, m/s).

This equation describes how energy increases as an object accelerates.

1.2. Mass-Energy Equivalence (Einstein's Equation)

Einstein’s famous equation describes the equivalence between mass and energy:

Where:

• \(E\) is the energy (in joules),
• \(m\) is the mass (in kilograms),
• \(c\) is the speed of light in a vacuum (\(c \approx 3.0 \times 10^8 \, \text{m/s}\)).

This equation tells us that a small amount of mass can be converted into a large amount of energy, which is key to understanding nuclear reactions.

1.3. Photon Energy

The energy of a photon, a particle of light, is given by:

Where:

• \(E_{\text{photon}}\) is the energy of the photon (in joules),
• \(h\) is Planck's constant (\(h \approx 6.626 \times 10^{-34} \, \text{J·s}\)),
• \(\nu\) is the frequency of the photon (in hertz, Hz).

Alternatively, the energy can also be expressed as a function of wavelength \(\lambda\):

Where:

• \(\lambda\) is the wavelength (in meters, m).

1.4. Relating Photon Energy to Mass

Though photons have no rest mass, they carry energy and can be associated with an effective relativistic mass using Einstein’s equation. The energy of a photon can be expressed as:

Where \(m_{\text{photon}}\) is the relativistic mass of the photon. Since photons move at the speed of light, their rest mass is zero, but they still possess energy and thus can be associated with an effective mass.

2. Derivations and Detailed Examples

2.1. Derivation of Kinetic Energy from Work

Kinetic energy can also be derived from the work-energy theorem, which states that the work done on an object is equal to the change in its kinetic energy. For a particle with mass \(m\) accelerating from rest under a force \(F\), the work done \(W\) is:

Using the kinematic equation \(v^2 = u^2 + 2 a d\) (where \(u = 0\) at rest), we can derive the kinetic energy formula:

2.2. Example: Calculating Photon Energy from Wavelength

Let’s calculate the energy of a photon with a wavelength of \(\lambda = 600 \, \text{nm}\) (which corresponds to yellow light).

Using the formula:

Substitute the known values:

This gives:

Thus, the energy of a photon of yellow light is approximately \(3.31 \times 10^{-19} \, \text{J}\).

2.3. Example: Energy and Mass of an Electron

The rest mass of an electron is approximately \(m_e = 9.11 \times 10^{-31} \, \text{kg}\). Using Einstein’s equation for mass-energy equivalence:

Substitute the known values:

This gives:

Thus, the energy equivalent of an electron’s rest mass is approximately \(8.2 \times 10^{-14} \, \text{J}\).

2.4. Example: Kinetic Energy of a Moving Car

Let’s calculate the kinetic energy of a car with mass \(m = 1500 \, \text{kg}\) moving at \(v = 30 \, \text{m/s}\). Using the kinetic energy formula:

Substitute the known values:

This gives:

Thus, the car’s kinetic energy is \(675000 \, \text{J}\) (or 675 kJ).