The energy of a particle can be described in several different ways, depending on its state of motion and the context in which it is considered. The most common forms of energy that particles can have are:
The kinetic energy (\(K\)) of a particle is the energy associated with its motion. The classical formula for kinetic energy is:
\[ K = \frac{1}{2}mv^2 \]
Where:
This formula is valid in classical mechanics, where the particle’s speed is much smaller than the speed of light (\(v \ll c\)). In this regime, the kinetic energy is proportional to the square of the velocity, meaning that doubling the speed of a particle quadruples its kinetic energy.
Let's calculate the kinetic energy of a particle with a mass of \(m = 2 \, \text{kg}\) and a velocity of \(v = 3 \, \text{m/s}\):
\[ K = \frac{1}{2} (2 \, \text{kg}) (3 \, \text{m/s})^2 = 9 \, \text{J} \]
The kinetic energy of this particle is \(9 \, \text{J}\) (joules).
Every particle has an intrinsic energy, even when it is at rest. This is known as the rest energy (\(E_0\)), and it is given by Einstein's famous equation:
\[ E_0 = mc^2 \]
Where:
Even if a particle is not moving, it still possesses energy due to its mass. This rest energy can be very large because of the large value of \(c^2\). For example, 1 kilogram of mass is equivalent to \(9 \times 10^{16} \, \text{J}\) of energy.
Let's calculate the rest energy of a particle with a mass of \(m = 1 \, \text{kg}\):
\[ E_0 = (1 \, \text{kg}) \times (3 \times 10^8 \, \text{m/s})^2 = 9 \times 10^{16} \, \text{J} \]
The rest energy of this particle is \(9 \times 10^{16} \, \text{J}\).
At high velocities (close to the speed of light), we must use the relativistic expression for total energy, which accounts for both the particle's rest energy and its kinetic energy. The total energy (\(E\)) of a particle is given by the equation:
\[ E^2 = (pc)^2 + (mc^2)^2 \]
Where:
If the particle is at rest (\(p = 0\)), this equation simplifies to:
\[ E = mc^2 \]
But if the particle is moving at high speeds, its total energy will include both the rest energy and the relativistic kinetic energy, which increases more than the classical formula suggests as the particle’s speed approaches the speed of light.
For a particle moving at relativistic speeds (close to the speed of light), the energy is calculated using the full relativistic formula. Suppose a particle has a rest mass of \(m = 1 \, \text{kg}\), and is moving with a velocity of \(v = 0.9c\) (90% of the speed of light). The relativistic momentum \(p\) is given by:
\[ p = \frac{mv}{\sqrt{1 - \left(\frac{v^2}{c^2}\right)}} \]
Substituting \(v = 0.9c\) into the formula:
\[ p = \frac{(1 \, \text{kg}) (0.9 \times 3 \times 10^8 \, \text{m/s})}{\sqrt{1 - (0.9)^2}} = 1.31 \times 10^{10} \, \text{kg} \cdot \text{m/s} \]
Now, using the total energy formula:
\[ E^2 = (1.31 \times 10^{10} \, \text{kg} \cdot \text{m/s} \times 3 \times 10^8 \, \text{m/s})^2 + (1 \, \text{kg} \times 3 \times 10^8 \, \text{m/s})^2 \]
Solving for \(E\), the total relativistic energy of this particle is much higher than its rest energy due to its high velocity.
Particles can have different forms of energy depending on their motion and state. The three main types of energy that particles exhibit are:
In everyday situations where velocities are much less than the speed of light, the classical formulas for kinetic energy and rest energy are sufficient. However, at high speeds, relativistic effects become significant, and the total energy of a particle must be calculated using the relativistic formula.