Momentum and energy are indeed related in the context of special relativity. In classical Newtonian physics, momentum (p) and kinetic energy (KE) are related for an object with mass (m) moving at non-relativistic speeds (much slower than the speed of light, c) by the equation:
KE = (1/2) * m * v^2 p = m * v
Where: KE = Kinetic energy m = Mass of the object v = Velocity of the object
However, in the theory of special relativity, which deals with objects moving at relativistic speeds (comparable to the speed of light), the relationship between momentum and energy becomes more intricate.
In special relativity, the correct expression for the energy of an object with mass is given by:
E = γ * m * c^2
Where: E = Energy of the object γ (gamma) = Lorentz factor, which is given by γ = 1 / sqrt(1 - v^2 / c^2) m = Mass of the object c = Speed of light in a vacuum
Notice that the kinetic energy formula used in classical physics, (1/2) * m * v^2, is only an approximation of the energy expression for low speeds when the Lorentz factor approaches 1.
To express momentum in special relativity, we use the concept of "relativistic momentum," which is given by:
p = γ * m * v
Comparing these expressions, we can see that momentum and energy are related in special relativity through the Lorentz factor (γ). As an object's speed approaches the speed of light (v ≈ c), the Lorentz factor approaches infinity, and both the momentum and energy of the object also increase significantly. This is why it becomes practically impossible to accelerate an object with mass to the speed of light, as it would require an infinite amount of energy.
In summary, momentum and energy are indeed related in special relativity, but the relationship is more complex than in classical physics. The Lorentz factor, which depends on the object's speed, plays a crucial role in connecting the two quantities and leads to interesting and counterintuitive effects at relativistic speeds.