Problem 11 - 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

(a) The new kinetic energy of the proton is twice the original: \( K_2 = 2K_1 \). (b) The energy of the second photon is twice the original: \( E_2 = 2E_1 \).

Step 1: Relate Kinetic Energy and Momentum of Proton

For a proton moving at a speed much slower than the speed of light, the kinetic energy \( K \) is related to momentum \( p \) by the formulas: \( K = \frac{p^2}{2m} \) and \( p = mv \), where \( m \) is the mass of the proton.Thus, \( K_1 = \frac{p_1^2}{2m} \) and when momentum is doubled \( p_2 = 2p_1 \), the new kinetic energy \( K_2 = \frac{p_2^2}{2m} = \frac{(2p_1)^2}{2m} = \frac{4p_1^2}{2m} = 2\frac{p_1^2}{2m} = 2K_1 \).

Step 2: Relate Energy and Momentum of Photon

For a photon, the relationship between energy \( E \) and momentum \( p \) is given by \( E = pc \), where \( c \) is the speed of light.For the first photon, \( E_1 = p_1c \).For the second photon with \( p_2 = 2p_1 \), the energy \( E_2 = p_2c = (2p_1)c = 2(p_1 c) = 2E_1 \).

Key Concepts

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. It is dependent on both the mass and speed of the object. When an object is moving, it has the ability to do work, which is the essence of kinetic energy. The formula to calculate kinetic energy for an object such as a proton moving at a speed much slower than light is given by: \[ K = \frac{p^2}{2m} \]Here, \( p \) represents the momentum of the object, and \( m \) is its mass. In the scenario where the momentum \( p \) is doubled, the kinetic energy becomes much larger. More specifically, if the momentum is doubled (\( p_2 = 2p_1 \)), the kinetic energy becomes:\[ K_2 = 2K_1 \]This is because squaring the doubled momentum results in a four-fold increase in the momentum term numerator, highlighting the quadratic relationship between momentum and kinetic energy.

Momentum

Momentum is a fundamental concept in classical mechanics, expressed as the product of an object's mass and velocity: \[ p = mv \]Momentum is a vector quantity, which means it has both a magnitude and a direction. It is crucial in analyzing the motion of objects, especially in collisions and other interactions. In simpler terms, momentum can be understood as the "quantity of motion" that an object has. If a proton has an increased momentum, this implies a change in either its mass (which remains constant for a single proton) or its velocity. Since the problem specifies much slower speeds than light, velocity changes are directly affecting the momentum. Doubling the momentum from \( p_1 \) to \( p_2 = 2p_1 \) signifies a need to analyze how this impacts kinetic energy, linking back to how energy grows quadratically with momentum changes.

Photon Energy

Photon energy is tied with the fundamental particle of light, the photon. Unlike objects with mass, photons are defined by their energy and momentum related via the speed of light. The energy of a photon can be expressed through the equation: \[ E = pc \]Here, \( E \) is the energy of the photon, \( p \) is its momentum, and \( c \) stands for the speed of light. This particular relationship highlights that as the momentum of a photon changes, its energy does too. In the exercise, if the momentum of another photon becomes double that of the initial photon (\( p_2 = 2p_1 \)), consequently, the energy also doubles, described as \( E_2 = 2E_1 \). This linear relationship contrasts with the quadratic relation of kinetic energy and demonstrates photons' unique properties due to their massless nature.

Speed of Light

The speed of light, denoted as \( c \), is one of the key constants in physics, at approximately \( 3 \times 10^8 \) meters per second. It acts as a fundamental bridge in the equations governing photon energy and momentum. Photons, being particles of light, always move at this speed in a vacuum, setting a universal speed limit. In our problem, both the kinetic energy of objects (via relativistic corrections) and photon energy are influenced by this constant. For photons, the equation linking energy and momentum, \( E = pc \), clearly showcases how intrinsic the speed of light is to energy calculations. Light's speed establishes the basis for much of modern physics, influencing concepts such as time dilation, relativity, and the interplay between mass and energy.