Problem 29 - University Physics with Modern Physics

Question

(a) At what speed is the momentum of a particle twice as great as the result obtained from the nonrelativistic expression \(m v ?\) Express your answer in terms of the speed of light. (b) A force is apphed to a particle along its direction of motion. At what speed is the magnitude of force required to produce a given acceleration twice as great as the force required to produce the same acceleration when the particle is at rest? Express your answer in terms of the speed of light.

Step-by-Step Solution

Answer

Both speeds are \( v = \frac{\sqrt{3}}{2}c \).

Step 1: Understanding Momentum in Classical and Relativistic Mechanics

In classical mechanics, the momentum of a particle is given by the expression \( p = mv \) where \( m \) is the mass and \( v \) is the velocity. In relativistic mechanics, the momentum is given by \( p = \gamma mv \), where \( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \) is the Lorentz factor. Our goal is to find the speed at which the relativistic momentum is twice the classical momentum.

Step 2: Equating Relativistic and Classical Momentum Expressions

We want \( \gamma mv = 2mv \). Simplifying, we get \( \gamma = 2 \). This means \( \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} = 2 \). Solving this equation for \( v \) will give us the speed where the relativistic momentum is twice the classical momentum.

Step 3: Solve for Speed using Lorentz Factor

We start from \( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} = 2 \). Squaring both sides, \( \frac{1}{1 - \frac{v^2}{c^2}} = 4 \). This leads to \( 1 - \frac{v^2}{c^2} = \frac{1}{4} \). Thus, \( \frac{v^2}{c^2} = 1 - \frac{1}{4} = \frac{3}{4} \). Solving for \( v \), we find \( v = \frac{\sqrt{3}}{2}c \).

Step 4: Understanding Force in Classical and Relativistic Contexts

In classical mechanics, force \( F \) is related to acceleration \( a \) by \( F = ma \). In relativistic dynamics, the force required to produce the same acceleration increases as the speed of the particle approaches the speed of light due to the increase in relativistic mass or effective inertia.

Step 5: Equating Relativistic and Classical Force Expressions

When asked to find when the force needed to produce a given acceleration is twice as much relativistically, we start with \( \gamma F = 2F = 2ma \), implying \( \gamma = 2 \). Just as earlier, we solve for the speed where this occurs.

Step 6: Calculate Speed using Lorentz Factor for Force

Following the process we derive for momentum, we again use \( \gamma = 2 \) and solve \( \frac{1}{1 - \frac{v^2}{c^2}} = 4 \) leading to \( v = \frac{\sqrt{3}}{2}c \). This is the same speed as when the momentum is doubled relativistically.

Key Concepts

Lorentz factor

In the realm of relativistic mechanics, the Lorentz factor plays a vital role in explaining how time, length, and physical mass are perceived differently when an object approaches the speed of light. The Lorentz factor, denoted as \( \gamma \), is given by the equation \( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \), where \( v \) is the velocity of the object, and \( c \) is the speed of light.

This factor becomes significant when speeds are comparable to the speed of light. At such speeds, time dilates, lengths contract, and mass effectively increases from the observer's frame of reference. The Lorentz factor is essentially a measure of these relativistic effects.

For example, in the problem regarding momentum, the task is to find when the relativistic momentum is twice the classical. Equating \( \gamma mv = 2mv \) confirms that \( \gamma \) must be 2. Solving for speed reveals that at \( v = \frac{\sqrt{3}}{2}c \), the Lorentz factor equals 2, highlighting significant relativistic effects.

momentum in physics

Momentum serves as a cornerstone concept in both classical and relativistic physics. Classically, momentum \( p \) is the product of an object's mass \( m \) and velocity \( v \), expressed as \( p = mv \). It reflects how much motion an object possesses and is conserved in isolated systems.

In relativistic physics, momentum is adjusted to account for how motion appears at speeds nearing the speed of light. Here, momentum adopts the form \( p = \gamma mv \), incorporating the Lorentz factor \( \gamma \). This adjustment ensures momentum conservation even when relativistic speeds are involved.

Understanding momentum's variation at different speeds is crucial. For instance, the exercise’s question asks for the speed causing relativistic momentum to be twice the classical momentum. By working through the Lorentz factor, it is discovered that this speed is \( v = \frac{\sqrt{3}}{2}c \), emphasizing that even though formulas may look similar, relativistic mechanics require deeper considerations.

classical vs relativistic dynamics

Classical and relativistic dynamics serve as two fundamental branches in physics, often intersecting when discussing motion, force, and energy.

**Classical Dynamics:**
We begin with classical dynamics, governed by Newton's laws, where force \( F \) is related to mass \( m \) and acceleration \( a \) by \( F = ma \). It effectively describes motion at everyday speeds and remains a practical model until relativistic velocities come into play.

**Relativistic Dynamics:**
At velocities approaching the speed of light, relativistic dynamics come into focus. They require adjusting classical equations to account for changing perceptions of space and time. For force, this means acknowledging that as speed increases towards light speed, more force is needed to achieve the same acceleration. This adjustment is due to the Lorentz factor, affecting how mass and force are realized in this framework.

In our specific example, the question about force shows that at \( v = \frac{\sqrt{3}}{2}c \), the needed relativistic force becomes twice the classical. Thus, while classical dynamics provide an excellent approximation for most situations, relativistic dynamics offer crucial insights when speeds tread near light speed.