How Does Relativistic Energy Affect Motion at High Speeds?
The conservation of energy is one of the most predominant laws in physics. We are aware of the fact that energy has many important forms, also each form can be converted to any other. We know that parameters like distance, time, motion, velocity and acceleration are all relativistic in nature, then, we can say that energy must be a relativistic quantity too. In other words, the energy of the object under consideration is depending on the inertial frame of reference that we are in.
According to classical physics, the total amount of energy in a system remains constant. Relativistically, energy is still conserved, provided its definition is moderated to include the possibility of mass changing into energy, as in the reactions that occur within a nuclear reactor. Relativistic energy is defined in such a way that it will be conserved in all inertial frames, just like in the case of relativistic momentum.
Relativistic Energy Formula:
Let us start with the derivation of the relativistic energy formula. When we are calculating the non-relativistic energy we assume that the change in kinetic energy is equal to the work done on the system or the object under consideration. In the relativistic case also assume the same theorem and estimate the relativistic energy.
To derive the relativistic energy formula we assume that the mass-energy principle holds good under relativity. According to the work-energy theorem, it states that the net work done on a system goes into kinetic energy. In other words, we say the change in kinetic energy can be evaluated by calculating the work done on the system or the object that we were considering. The relativistic energy formula is also known as the energy-momentum relation.
In classical physics, the kinetic energy is given by the product of mass and square of the velocity, this kinetic energy is valid for the object which has a velocity less than the speed of light. But we know that in relativistic mechanics we assume that the particles are moving with the speed of light. The potential energy of the particle is considered to be almost zero or negligible.
The relativistic energy is also known as the relativistic kinetic energy and it can be derived by a small derivation as given below. We know that according to the work-energy theorem, it states that the net work done on a system goes into kinetic energy. In other words, we say the change in kinetic energy can be evaluated by calculating the work done on the system or the object that we were considering. Therefore mathematically we write,
\[\Rightarrow E=\int_{0}^{r}dW=\int_{0}^{r}F.dr=\int_{0}^{r}Fdr\]…….(1)
Since both applied force and the displacement are in the same direction we are not considering the scalar product.
We know that according to Newton’s law of motion force F is equal to,
\[\Rightarrow F=\frac{dp}{dt}=\frac{d(mv)}{dt}\]…….(2)
And the velocity of the particle is given by,
\[\Rightarrow v=\frac{dr}{dt}\]…………(3)
Substituting the values in the equation (1) we get,
\[\Rightarrow E=\int_{0}^{t}\frac{d}{dt}(mv)vdt=m\int_{0}^{v}vdv=\frac{mv^{2}}{2}\]..(4)
Equation (4) resembles the classical kinetic energy formula of the particle moving with velocity v. But, here we are considering the relativistic case and hence substituting the value of mass by rest mass and velocity by Lorentz transformation we get,
\[\Rightarrow m=\gamma m_{0}c^{2}\]
Substituting the value of relativistic mass and velocity we finally calculate the relativistic energy.
FAQs on Relativistic Energy in Physics: Explanation & Applications
1. What is relativistic energy in simple terms?
Relativistic energy is the total energy an object has when its motion is considered according to Einstein's theory of relativity. It's especially important for objects moving at speeds close to the speed of light. This total energy is made up of two parts: its rest energy (the energy it has just from its mass) and its kinetic energy (the energy of its motion).
2. What is the formula for total relativistic energy?
The formula for total relativistic energy is E = γmc². Here's what each part means:
- E is the total relativistic energy.
- γ (gamma) is the Lorentz factor, which increases as the object's speed increases.
- m is the object's rest mass (its mass when it's not moving).
- c is the constant speed of light.
3. What is the main difference between relativistic and non-relativistic energy?
The key difference is how they treat mass. In classical (non-relativistic) physics, an object's mass is always constant. In relativistic physics, an object's effective mass (or inertia) increases as it gets faster. This means its total energy grows much more significantly at high speeds than classical formulas would predict.
4. Can you give an example of relativistic energy in the real world?
A perfect example is a nuclear power plant. In nuclear fission, a small amount of mass from atoms is converted into a very large amount of energy. This direct conversion of mass to energy is explained by the rest energy formula, E=mc², which is a fundamental part of the concept of relativistic energy.
5. What does 'rest energy' (E₀ = mc²) actually mean? Does a stationary object have energy?
Yes, 'rest energy' means that mass itself is a concentrated form of energy. Even an object that is completely still has a huge amount of energy locked inside its mass. The famous formula E₀ = mc² tells us exactly how much energy is equivalent to its rest mass. This is the energy that is released in nuclear reactions.
6. How is relativistic kinetic energy different from the classical formula (1/2 mv²)?
The classical formula, K.E. = 1/2 mv², is an excellent approximation for everyday, low speeds. However, it becomes inaccurate as an object approaches the speed of light. Relativistic kinetic energy is more precise and is calculated as K.E. = (γ - 1)mc². At very high speeds, this formula shows that the kinetic energy required for acceleration becomes almost infinite, which the classical formula fails to predict.
7. Why can't an object with mass ever reach the speed of light?
This is because of the Lorentz factor (γ) in the energy formula E = γmc². As an object's velocity gets closer to the speed of light, the Lorentz factor approaches infinity. This means it would take an infinite amount of energy to accelerate the object to the speed of light. Since an infinite amount of energy is not available, reaching this speed is physically impossible for any object with mass.
