Problem 20 - College Physics

Question

\(\bullet\) 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 \(0.650 c,\) and the speed of each particle relative to the other is 0.950\(c .\) 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 approximately 0.784c.

Step 1: Understand the Problem

We have two particles, and we know the speed of the first particle is \(0.650c\) relative to a stationary laboratory observer. We need to find the speed of the second particle as measured in the laboratory, given that the speed of the second particle relative to the first is \(0.950c\).

Step 2: Use the Relativistic Velocity Addition Formula

The relativistic velocity addition formula is given by: \[ u' = \frac{u + v}{1 + \frac{uv}{c^2}} \] where \(u\) and \(v\) are the velocities of the two objects relative to an observer, and \(u'\) is the velocity of one object relative to the other. In our problem, \(u = 0.650c\) and \(u' = 0.950c\). We need to solve for \(v\), which is the velocity of the second particle relative to the laboratory.

Step 3: Solve for the Velocity of the Second Particle

Plug in the known values into the relativistic velocity addition formula: \[ 0.950c = \frac{0.650c + v}{1 + \frac{0.650c \cdot v}{c^2}} \] Simplify and solve for \(v\):Simplify left side:\[ 0.950 = \frac{0.650 + v/c}{1 + 0.650v/c^2} \]Cross-multiply:\[ 0.950 (1 + 0.650v/c^2) = 0.650 + v/c \]Distribute on left:\[ 0.950 + 0.6175v/c^2 = 0.650 + v/c \]Rearrange terms to collect v (replace \(v\) with \(vc\) to make steps clearer):\[ 0.950 - 0.650 = (v/c) - 0.6175 (v/c^2) \]\[ 0.300 = (v/c)(1 - 0.6175/c) \]Solve for \(v\):\[ v = \frac{0.300c}{1 - 0.6175} \]Calculate:\[ v = \frac{0.300c}{0.3825} \approx 0.784c \]

Step 4: Conclusion

The velocity of the second particle, as measured in the laboratory, is approximately \(0.784c\).

Key Concepts

Special Relativity

Special relativity is a fascinating theory introduced by Albert Einstein, which fundamentally changed how we understand the physics of high-speed travel and motion. One of its principal ideas is that the laws of physics are the same for all observers, regardless of their constant velocity relative to each other. This leads to surprising results, particularly when dealing with objects moving at velocities close to the speed of light, denoted by \( c \).

A critical aspect of special relativity is its effect on time and space. As an object's speed approaches the speed of light, time appears to slow down for it relative to a stationary observer, an effect known as time dilation. Simultaneously, distances appear contracted along the direction of motion, called length contraction. These effects ensure that no object with mass can reach or exceed the speed of light.

The concept of relativistic velocity addition also arises from special relativity. Unlike classical physics, where velocities simply add up, special relativity uses a specific formula that takes into account the finite and invariant nature of the speed of light, ensuring the resulting velocity doesn't exceed \( c \). This ensures that observers in different reference frames measure velocities relative to each other in a way that's consistent with relativity's principles.

Particle Physics

Particle physics is the branch of science that explores the smallest constituents of matter and the fundamental forces governing them. In high-energy accelerators, particles are often sped up to velocities near the speed of light. Understanding their velocities becomes essential, particularly when they collide or move in opposite directions, as described in our exercise.

These high-speed experiments allow researchers to investigate the interactions of particles that make up the universe, such as protons, neutrons, and electrons. Scientists use this information to test predictions of the Standard Model, the prevailing theory describing particle interactions. Notably, relativistic effects, like those in special relativity, are crucial for interpreting these high-speed interactions accurately.

By analyzing the results of such experiments, physicists can gather insights into phenomena like antimatter, the nature of neutrinos, and even explore beyond the Standard Model physics. This aids in understanding both the fundamental particles and the forces that hold matter together, providing clues to the fabric of our universe.

Velocity Calculations

Velocity calculations in the context of special relativity differ significantly from those in classical mechanics. The relativistic velocity addition formula is a key tool in these calculations, used to determine how speeds add up when objects move at relativistic speeds, that is, speeds close to the speed of light.

The formula for relativistic velocity addition is:

• \( u' = \frac{u + v}{1 + \frac{uv}{c^2}} \)

Where \( u \) is the velocity of one object relative to a stationary observer, \( v \) is the velocity of the other object, and \( u' \) is the velocity of one object relative to the other. This formula ensures that the resulting velocity \( u' \) never exceeds the speed of light \( c \).

In our exercise, this formula helps us find out how fast the second particle is moving relative to the laboratory frame, even when both particles are moving close to the speed of light. Calculations involve plugging in the given velocities, manipulating the equation algebraically, and solving for the unknown velocity. Understanding and applying these calculations are fundamental when discussing the behavior of particles moving at such high speeds.