Problem 19 - University Physics with Modern Physics

Question

Two particles are created in a high-energy accelerator and move off in opposite directions. The speed of one particle, as measured in the laboratory, is 0.650 \(\mathrm{c}\) , and the speed of each partcle relative to the other is 0.950 \(\mathrm{c}\) . What is the speed of the second particle, as measured in the laboratory?

Step-by-Step Solution

Answer

The speed of the second particle is 0.784c.

Step 1: Identify Given Variables

You have two speeds given in the problem. Let the speed of particle 1 as measured in the laboratory be \( v_1 = 0.650 \mathrm{c} \). The relative speed of one particle to another is given as \( v_{rel} = 0.950 \mathrm{c} \). We need to find the speed of particle 2 in the laboratory, which we will call \( v_2 \).

Step 2: Use the Velocity Addition Formula

To find the speed of the second particle, we apply the relativistic velocity addition formula: \[ v_{rel} = \frac{v_1 + v_2}{1 + \frac{v_1 v_2}{c^2}}. \] This formula helps determine the relative velocity in a system considering relativistic effects.

Step 3: Substitute Known Values

Substitute \( v_{rel} = 0.950c \), \( v_1 = 0.650c \), into the formula: \[ 0.950c = \frac{0.650c + v_2}{1 + \frac{0.650c \cdot v_2}{c^2}}. \] Simplify by dividing by \( c \) to get: \[ 0.950 = \frac{0.650 + v_2}{1 + 0.650v_2}. \]

Step 4: Solve for \( v_2 \)

Rearrange the equation to solve for \( v_2 \): \[ 0.950(1 + 0.650v_2) = 0.650 + v_2. \] Expand and simplify: \[ 0.950 + 0.6175v_2 = 0.650 + v_2. \] Rearrange to get: \[ 0.950 - 0.650 = v_2 - 0.6175v_2. \] Simplify: \[ 0.300 = 0.3825v_2. \] Solve for \( v_2 \): \[ v_2 = \frac{0.300}{0.3825} \approx 0.784 \mathrm{c}. \]

Step 5: Verify Solution

Ensure no inaccuracies were present in substitutions and calculations. The computed speed of particle 2 in the laboratory \( v_2 = 0.784 \mathrm{c} \) seems reasonable considering relativistic velocity transformations.

Key Concepts

Relativity

Relativity is a fundamental concept in physics that describes how the laws of physics apply to different observers moving relative to each other. Albert Einstein's theory of relativity is divided into two parts: special relativity and general relativity. In this context, we focus on special relativity, which primarily deals with objects moving at constant velocities, particularly those moving at speeds close to the speed of light (denoted as \(c\)).

Special relativity introduces two key ideas:

• Time Dilation: Time intervals are relative and can differ for observers in different inertial frames.
• Length Contraction: Objects appear shorter in the direction of motion for an observer moving relative to the object.

These ideas lead us to understand phenomena such as relativistic velocity addition, where velocities do not simply add up as they do in classical physics.

Relativity alters our classical notions of space and time, merging them into a single entity called spacetime. It guides how we solve problems that involve high velocities, like the one in the exercise, ensuring that solutions are consistent with the principles of nature.

Velocity Transformations

Velocity transformations are a crucial aspect of understanding motion in the context of relativity. Classical physics used simple addition to combine velocities, but this approach fails at speeds close to \(c\), the speed of light. The relativistic velocity addition formula is used to calculate velocities in such scenarios:

\[ v_{rel} = \frac{v_1 + v_2}{1 + \frac{v_1 v_2}{c^2}} \]

This formula helps us find the relative velocity \( v_{rel} \) between two objects that are in motion. When speeds approach the speed of light, relativistic effects become significant, and this formula corrects for those effects, ensuring computations keep in line with special relativity.

Key Points for Velocity Transformations:

• The relativistic velocity addition formula demonstrates that velocities do not simply add in a linear fashion as one might expect.
• It accounts for the effects of time dilation and length contraction.
• This formula is critical in particle physics where particles often move at speeds approaching \(c\).

Understanding this principle assists in comprehending why certain calculations, such as in the given exercise, yield results that deviate from classical predictions.

Particle Physics

Particle physics is a branch of physics that explores the fundamental constituents of matter and their interactions. It seeks to unravel the complexity of particles beyond atoms, dealing with entities like quarks, leptons, and bosons.

High-energy accelerators are essential tools in particle physics, allowing scientists to create and study particles moving at relativistic speeds. In the original exercise, particles are created in a high-energy accelerator moving at speeds significant enough to require relativistic calculations.

Applications and Importance:

• High-energy accelerators enable the exploration of subatomic particle behavior at energy levels that mimic conditions just after the Big Bang.
• Relativistic calculations allow scientists to predict and verify particle properties and interactions accurately.
• Insights from particle physics contribute to a deeper understanding of the universe's fundamental workings, impacting fields like cosmology and quantum mechanics.

Applying relativistic velocity transformations in particle physics is crucial for correctly interpreting experimental results, such as those derived from particle collisions at near-light speeds. This understanding broadens our grasp of fundamental forces and particles at the most elemental levels of matter.