Problem 85 - Fundamentals of Physics

Question

One cosmic-ray particle approaches Earth along Earth's north-south axis with a speed of \(0.80 \mathrm{c}\) toward the geographic north pole, and another approaches with a speed of \(0.60 c\) toward the geographic south pole (Fig. 37 - 34). What is the relative speed of approach of one particle with respect to the other?

Step-by-Step Solution

Answer

The relative speed is approximately \(0.946c\).

Step 1: Understand the Problem

We need to find the relative speed between two particles approaching each other along the north-south axis of the Earth. One particle travels at a speed of \(0.80c\) toward the north pole and the other at \(0.60c\) toward the south pole, where \(c\) is the speed of light.

Step 2: Relative Speed Formula

To find the relative speed of two objects moving towards each other at relativistic speeds, we use the formula: \[v = \frac{u + v'}{1 + \frac{uv'}{c^2}}\]where \(u\) and \(v'\) are the speeds of the particles (\(0.80c\) and \(0.60c\)), and \(v\) is the relative speed.

Step 3: Plug in the Values

Substitute \(u = 0.80c\) and \(v' = 0.60c\) into the formula: \[v = \frac{0.80c + 0.60c}{1 + \frac{(0.80c)(0.60c)}{c^2}}\]

Step 4: Simplifying the Denominator

Simplify the denominator \(1 + \frac{(0.80)(0.60)c^2}{c^2} = 1 + 0.48 = 1.48\).

Step 5: Calculate the Relative Speed

Substitute into the equation: \[v = \frac{1.40c}{1.48}\]Calculate \(v\): \[v \approx 0.946c\]

Step 6: Conclude the Result

The relative speed of approach of one cosmic-ray particle with respect to the other is approximately \(0.946c\).

Key Concepts

Cosmic-ray Particle

Cosmic-ray particles are high-energy particles that originate from outer space. They travel at nearly the speed of light and consist mostly of protons and atomic nuclei. When these particles enter the Earth's atmosphere, they can create showers of secondary particles. Understanding cosmic-ray particles is essential in physics for various reasons:

• They help in studying particle interactions at high energies.
• They provide insight into processes happening in stellar environments.
• They are relevant for radiation protection in space travel.

Despite being highly energetic, the exact origin of cosmic-ray particles is not fully understood. Possible sources include supernova remnants, the Sun, or even distant galaxies.

Relative Speed

In physics, relative speed is the speed of one object as observed from another object. For everyday speeds, this is simply the sum of their individual speeds if they are moving toward each other. However, this isn't true at relativistic speeds, or speeds close to that of light.

When particles approach relativistic speeds, we can't just add their speeds. Doing so would imply a speed greater than the speed of light, which is impossible according to the principles of special relativity. This is why we use a specific formula to calculate relative speeds:
\[ v = \frac{u + v'}{1 + \frac{uv'}{c^2}} \]Here, \(u\) and \(v'\) are the speeds of the particles, and \(c\) represents the speed of light.

Special Relativity

Special relativity, introduced by Albert Einstein in 1905, revolutionized our understanding of time, space, and motion. It primarily deals with objects moving at high speeds, approaching the speed of light. The theory is based on two main postulates:

• The laws of physics are the same for all observers in uniform motion relative to one another.
• The speed of light in a vacuum is constant and will be the same for all observers, regardless of the motion of the light source or observer.

These postulates lead to fascinating consequences:

• Time dilation: Time moves slower for objects moving at high speeds.
• Length contraction: Objects moving at speeds close to light appear shorter in the direction of motion.
• Mass-energy equivalence: As expressed in the formula \(E = mc^2\), showing that energy and mass are equivalent.

Special relativity ensures that no object with mass can move faster than the speed of light, a fundamental tenet in physics.

Speed of Light

The speed of light, denoted as \(c\), is a universal constant in physics with a value of approximately \(299,792,458\) meters per second. It represents the maximum speed at which all energy, matter, and information in the universe can travel.

Light is composed of photons, massless particles that always move at this speed in a vacuum. This speed serves as a critical limit:

• No object with mass can reach or exceed the speed of light.
• It's the foundation of the theory of relativity, affecting time and space measurements.

Understanding the speed of light is crucial to advanced concepts in physics, and it allows scientists to make accurate predictions about the behavior of particles at high velocities.