Question

A photon and an object of mass \(m\) have the same momentum \(p\). (a) Assuming that the massive object is moving slowly, so that nonrelativistic fonnulas are valid. find in terms of \(m, p\), and \(c\) the ratio of the massive ob)ect's kinetic energy to the photon's kinetic ener \(g y\), and argue that it is small. (b) Find the same ratio found in part (a), but using relativistically correct formulas for the massive object. (Nore: \(E^{2}=p^{2} c^{2}+m^{2} c^{4}\) may be helpful.) (c) Show that the low-speed limit of the ratio of part (b) agrees with part (a) and that the high-speed limit is \(1 .\) (d) Show that at very high speod, the kinetic energy of a massive object approaches \(\rho \mathrm{c}\),

Short answer

The non-relativistic ratio of kinetic energies is \(0.5*p/mc\), which is small. The relativistic ratio is \(sqrt((p/mc)^2 + 1) - 1\), which equals the non-relativistic ratio for the low-speed limit and approaches 1 for the high-speed limit. At very high speed, the kinetic energy of a massive object approaches \(pc\).

Step-by-step solution

Eq for nonrelativistic kinetic energy

The nonrelativistic equation for kinetic energy is \( KE = 0.5*m*v^2 \). Because in this scenario, the momentum \( p \) of the massive object equals the momentum of the photon \( p = E/c \) (where \( E \) is the energy of the photon and \( c \) is the speed of light), we can establish that \( v = p/m \). By substitution we get \( KE = 0.5*p^2/m \).

Photon Kinetic energy

The kinetic energy \( E \) for a photon is \( E = pc \). So the ratio of the massive object's kinetic energy to the photon's kinetic energy is \( 0.5*p^2/m / (pc) = 0.5*p/mc \). Because \( m >> p/c \), this ratio is small, as argued in the problem statement.

Relativistic Kinetic Energy for Massive Object

Relativistically, the total energy of the massive object is given by \( E^2 = p^2c^2 + m^2c^4 \), from which we get \( E = sqrt(p^2c^2 + m^2c^4) \). The kinetic energy is the total energy minus the rest-mass energy, i.e., \( KE = E - mc^2 = sqrt(p^2c^2 + m^2c^4) - mc^2 \).

Ratio in Relativistic Case

The ratio from part (b) is \( KE/E = (sqrt(p^2c^2 + m^2c^4) - mc^2)/(pc) \), which simplifies to \( sqrt((p/mc)^2 + 1) - 1 \). This equals the ratio from part (a) for the low-speed limit, when \( p/mc << 1 \). And for the high-speed limit, we see that as \( p/mc \) gets much larger than \( 1 \), this ratio approaches \( 1 \).

Kinetic energy at high speed

At very high speed, the kinetic energy of a massive object approaches its momentum times the speed of light, \( KE = pc \)

Key concepts

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion. In classical mechanics, for an object of mass \( m \) moving at a velocity \( v \), the kinetic energy is given by the formula \( KE = \frac{1}{2} m v^2 \). This formula works perfectly when dealing with speeds much smaller than the speed of light. However, when speeds approach the speed of light, relativistic effects become significant, and we need to use relativistic equations.

For a massive object moving at relativistic speeds, the kinetic energy formula is adjusted to account for relativistic effects. It is calculated by subtracting the rest mass energy from the total energy of the object. This is given by:

• \( E = \sqrt{p^2 c^2 + m^2 c^4} \)
• \( KE = E - m c^2 \)

The rest mass energy \( m c^2 \) is the energy an object possesses if it were at rest, and this needs to be subtracted from the total energy to get the kinetic energy.

In comparing kinetic energies of different objects, such as photons and massive objects, it's crucial to use the correct formulas to account for their distinct behaviors at high speeds.

Photon Momentum

Photon momentum might seem counterintuitive since photons don’t have mass in the traditional sense. However, they do possess momentum, which is a fundamental property of all particles with energy. For a photon, its energy \( E \) is related to its momentum \( p \) by the equation \( E = pc \), where \( c \) is the speed of light.

This relationship is a cornerstone of understanding photon behavior in the context of relativistic physics. It implies that the momentum of a photon is directly proportional to its energy and inversely proportional to the speed of light. Because of this property:

• Photons can exert pressure (called radiation pressure) on surfaces they collide with.
• They can transfer energy and momentum in interactions, such as in the photoelectric effect.

The concept of photon momentum becomes even more interesting as speeds increase in systems involving massive objects. In high-speed scenarios, photon momentum demonstrates how energy and momentum maintain their significant influence even without mass.

Relativity

The theory of relativity, developed by Albert Einstein, revolutionizes our understanding of physics by explaining the interrelation of space, time, and motion. It includes both the special and general theories. A core component of special relativity is the inability of objects with mass to reach the speed of light and the different way we must treat physical quantities as things approach relativistic speeds.

In the realm of relativity, we move beyond Newtonian concepts to describe how time and space are linked and how the laws of physics apply at high velocities. Key insights include:

• The speed of light is the ultimate speed limit in the universe.
• Mass and energy are equivalent, as expressed in the famous equation \( E = mc^2 \).
• As objects move faster, their mass effectively increases, affecting how we calculate their kinetic energy and other related phenomena.

Understanding relativity allows us to predict how objects behave when moving at significant fractions of the speed of light. It gives us the frameworks needed to precisely describe phenomena such as time dilation, length contraction, and energy-momentum dynamics.