Problem 1 - University Physics with Modern Physics

Question

(a) A proton is moving at a speed much slower than the speed of light. It has kinetic energy \(K_{1}\) and momentum \(p_{1} .\) If the momentum of the proton is doubled, so \(p_{2}=2 p_{1},\) how is its new kinetic energy \(K_{2}\) related to \(K_{1} ?\) (b) A photon with energy \(E_{1}\) has momentum \(p_{1} .\) If another photon has momentum \(p_{2}\) that is twice \(p_{1},\) how is the energy \(E_{2}\) of the second photon related to \(E_{1} ?\)

Step-by-Step Solution

Answer

Both the proton's new kinetic energy and the photon's new energy are twice their initial values.

Step 1: Analyze the Proton's Initial Conditions

For a proton moving much slower than the speed of light, the kinetic energy can be expressed as \( K_1 = \frac{p_1^2}{2m} \), where \( m \) is the mass of the proton. The initial momentum is \( p_1 \).

Step 2: Find the Proton's New Kinetic Energy

If the momentum is doubled, \( p_2 = 2p_1 \). The new kinetic energy is \( K_2 = \frac{p_2^2}{2m} = \frac{(2p_1)^2}{2m} = \frac{4p_1^2}{2m} = 2 \cdot \frac{p_1^2}{2m} = 2K_1 \). Thus, the new kinetic energy is twice the initial kinetic energy.

Step 3: Understand the Photon-P's Relationship

For a photon, the relationship between energy and momentum is given by \( E = pc \), where \( c \) is the speed of light. Initially, the photon has energy \( E_1 = p_1c \).

Step 4: Determine the Second Photon's Energy

If the second photon's momentum is \( p_2 = 2p_1 \), its energy is \( E_2 = p_2c = (2p_1)c = 2p_1c = 2E_1 \). Therefore, the energy of the second photon is twice the energy of the first photon.

Key Concepts

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion. It's a core concept in physics, particularly in classical mechanics. For objects moving at speeds much slower than the speed of light, kinetic energy \[ K = \frac{1}{2}mv^2 \] where \( m \) is the mass of the object and \( v \) is its velocity. Another useful expression for kinetic energy when considering momentum is \[ K = \frac{p^2}{2m} \] where \( p \) is momentum. In the given problem, when the momentum of a proton doubles, its kinetic energy also changes. Using the momentum-based expression, if the proton's momentum is initially \( p_1 \), with kinetic energy \( K_1 = \frac{p_1^2}{2m} \), and then its momentum is doubled \[ p_2 = 2p_1 \] so the new kinetic energy becomes \[ K_2 = \frac{(2p_1)^2}{2m} = 2K_1 \] Hence, the kinetic energy doubles, showcasing how sensitive it is to changes in momentum values.

Momentum

Momentum is a fundamental property in physics that describes the quantity of motion an object possesses. It's given by the product of an object's mass and velocity: \[ p = mv \] This means that both the speed and mass of an object are integral in determining its momentum. In classical mechanics, momentum is conserved in isolated systems, which makes it particularly useful in analyzing collisions and interactions.

• Momentum is directional; it has both magnitude and direction.
• It is a vector quantity.

In the given scenario, doubling the proton's momentum implies a direct modification in its speed while keeping its mass constant. This directly influences the kinetic energy, as seen in kinetic energy equations that involve momentum.

Photon Energy

Photon energy is a special area of study since photons are particles of light with zero rest mass, moving at the speed of light, \( c \). The energy of a photon relates directly to its momentum via the equation \[ E = pc \] where \( E \) is the energy, \( p \) is the momentum, and \( c \) is the speed of light. This is a manifestation of wave-particle duality in quantum mechanics and is crucial in understanding phenomena in optics and electromagnetic radiation. In the exercise, one photon has energy \( E_1 = p_1c \), and when its momentum is doubled \( p_2 = 2p_1 \), the energy becomes \[ E_2 = 2p_1c = 2E_1 \] This demonstrates that energy is directly proportional to momentum for photons. Doubling momentum results in doubling energy, making energy manipulation very elegant in light physics.

Relativistic Physics

Relativistic physics deals with phenomena occurring at speeds close to the speed of light. In this realm, traditional Newtonian mechanics no longer apply, and we must use the principles of Einstein's theory of relativity. Key differences include how time and space are perceived, and the relationships between energy, momentum, and mass.

• Mass increases with velocity in relativistic speeds.
• Time dilation and length contraction occur.
• Newton's second law is modified for relativistic speeds.

For particles much slower than the speed of light, like the proton in our problem, relativistic effects are negligible. However, photons are relativistic particles as they always move at the speed of light. Their energy and momentum relations reflect that. Understanding these principles is essential for fields like quantum mechanics and cosmology, where high-speed dynamics dominate.