# Kinetic Energy Dependent on Frame of Reference?

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• UMath1
In summary: You can write it as a matrix exponential ##{\Lambda^{\mu}}_{\nu}=\exp\left(\frac{\mathrm{d} x^{\mu}}{\mathrm{d} x'^{\nu}}\right)## of the four-velocity ##\mathrm{d} x^{\mu}/\mathrm{d} \tau##.In summary, according to special relativity, kinetic energy is frame dependent and given by the equation KE= ([PLAIN]https://upload.wikimedia.org/math/3/3/4/334de1ea38b615839e4ee6b65ee1b103.png-1)(mv^2)/2. The rest energy is a relativistic

#### UMath1

According to special relativity, KE= ([PLAIN]https://upload.wikimedia.org/math/3/3/4/334de1ea38b615839e4ee6b65ee1b103.png-1)(mv^2)/2. [Broken] Since the velocity measured is dependent on a person's frame of reference, then does that mean energy too is dependent on frame of reference? For example, if an observer in Car A is moving at velocity, V, and a person in Car B is moving at velocity, V, also, then according to the observer in Car A, the person in Car B would have 0 J of kinetic energy?

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Kinetic energy is indeed frame dependent.

UMath1 said:
According to special relativity, KE= ([PLAIN]https://upload.wikimedia.org/math/3/3/4/334de1ea38b615839e4ee6b65ee1b103.png-1)(mv^2)/2. [Broken] Since the velocity measured is dependent on a person's frame of reference, then does that mean energy too is dependent on frame of reference? For example, if an observer in Car A is moving at velocity, V, and a person in Car B is moving at velocity, V, also, then according to the observer in Car A, the person in Car B would have 0 J of kinetic energy?
Energy is dependent on the frame of reference already in classical mechanics. In the rest frame of any object, the object has zero kinetic energy.

The kinetic energy in special relativity is given by ##T = (\gamma - 1) m c^2##, nothing else. For small ##v##, this is well approximated by the classical expression ##T = mv^2/2##.

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UMath1 said:
Since the velocity measured is dependent on a person's frame of reference, then does that mean energy too is dependent on frame of reference?

Yes. This is also true in Newtonian physics.

In special relativity kinetic energy is ##\gamma mc^2-mc^2##. The first term is the total energy, it's frame dependent. The second term is the rest energy, it's a relativistic invariant (has the same value in all reference frames).

One should note that the four-momentum
$$(p^{\mu})=\begin{pmatrix} E/c \\ \vec{p} \end{pmatrix}$$
is a four-vector if you define the energy such that it includes the rest energy, i.e., ##E=E_0+E_{\text{kin}}=c \sqrt{m^2 c^2+\vec{p}^2}##. Then the covariant expression
$$p_{\mu} p^{\mu}=m^2 c^2$$
contains the relation between energy and momentum.

In terms of space-time coordinates along the particle's trajectory you have
$$p^{\mu}=m \frac{\mathrm{d} x^{\mu}}{\mathrm{d} \tau} = m \frac{\mathrm{d} x^{\mu}}{\mathrm{d} t} \frac{\mathrm{d} t}{\mathrm{d} \tau} = m \gamma \begin{pmatrix} c \\ \mathrm{d} x/\mathrm{d} t \end{pmatrix}.$$
Thus the energy and momentum in terms of the three-velocity is
$$E=c p^0=m \gamma c^2, \quad \vec{p}=m \gamma \vec{v} \; \Rightarrow \; \vec{\beta}=\frac{\vec{v}}{c}=\frac{c \vec{p}}{E}.$$
If you need to transform from one to another frame via a Lorentz transformation, use covariant quantities like the four-momentum and then the just given expressions to derive the three velocity and other non-covariant quantity with its components in the new frame. That makes things much more convenient and much less dangerous to make a mistake. As any four-vector the four-momentum transforms as
$$p^{\prime \mu}={\Lambda^{\mu}}_{\nu} p^{\nu},$$
where ##{\Lambda^{\mu}}_{\nu}## is the Lorentz-transformation matrix.

## 1. What is meant by "frame of reference" in terms of kinetic energy?

Frame of reference refers to the point from which we are observing the motion of an object. It is the perspective or viewpoint that we use to describe the motion of an object.

## 2. How does kinetic energy change depending on the frame of reference?

Kinetic energy is a relative quantity and depends on the frame of reference. This means that the value of kinetic energy will change depending on the observer's point of view. For example, an object may have a different kinetic energy when observed from a stationary frame of reference compared to when observed from a moving frame of reference.

## 3. Can an object have different kinetic energies in different frames of reference?

Yes, an object can have different kinetic energies in different frames of reference. This is because kinetic energy is dependent on both the mass and velocity of an object, and these values can vary depending on the observer's frame of reference.

## 4. How does the concept of relativity relate to kinetic energy and frame of reference?

The concept of relativity states that the laws of physics are the same in all inertial frames of reference. This means that the kinetic energy of an object will be the same in all inertial frames of reference, even though the actual values may differ.

## 5. Are there any real-world applications of understanding kinetic energy in different frames of reference?

Yes, understanding how kinetic energy changes depending on the frame of reference is important in many fields, such as engineering, physics, and astronomy. For example, in engineering, it is essential to consider the frame of reference when designing structures that will experience different forces and energies in different frames of reference.