Question

Two particles in a high-energy accelerator experiment approach each other head-on with a relative speed of 0.890c. Both particles travel at the same speed as measured in the laboratory. What is the speed of each particle, as measured in the laboratory?

Short answer

Each particle's speed in the laboratory is approximately 0.919c.

Step-by-step solution

Understand the Problem

We are asked to find the speed of each particle relative to the laboratory, given that they have a relative speed of 0.890c as they approach head-on.

Use the Relativistic Velocity Addition Formula

Relativistic velocity addition helps us calculate the laboratory speed since the speeds are close to the speed of light. The formula is:\[v = \frac{v_1 + v_2}{1 + \frac{v_1 v_2}{c^2}}.\] In this case, the relative speed \(v_{rel} = 0.890c\), and since \(v_1 = v\) and \(v_2 = v\), the formula becomes:\[0.890c = \frac{2v}{1 + \frac{v^2}{c^2}}.\]

Solve for Each Particle's Speed

Substitute \(v_{rel} = 0.890c\) into the formula from the previous step and solve for \(v\):\[0.890c = \frac{2v}{1 + \frac{v^2}{c^2}} \Rightarrow 0.890 = \frac{2v/c}{1 + v^2/c^2}.\]Multiply both sides by \(1 + v^2/c^2\):\[0.890(1 + v^2/c^2) = 2v/c.\]Expand and rearrange:\[0.890 + 0.890 \frac{v^2}{c^2} = 2 \frac{v}{c}.\]Multiply through by \(c\) to remove fractions:\[0.890c + 0.890 \frac{v^2}{c} = 2v.\]

Simplify and Solve Quadratic Equation

Rearrange the previous equation to form a quadratic equation in terms of \(v\):\[0.890 v^2 - 2cv + 0.890c^2 = 0.\]Apply the quadratic formula \(v = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 0.890, b = -2c, c = 0.890c^2\):\[v = \frac{2c \pm \sqrt{(-2c)^2 - 4(0.890)(0.890c^2)}}{2(0.890)}.\]

Calculate the Solution

Simplify further:\[v = \frac{2c \pm \sqrt{4c^2 - 3.1684c^2}}{1.78}.\]\[v = \frac{2c \pm \sqrt{0.8316c^2}}{1.78}.\]\[v = \frac{2c \pm 0.9119c}{1.78}.\]Choose the positive root for the speed:\[v = \frac{2c + 0.9119c}{1.78} = \frac{2.9119c}{1.78} = 1.6353c/1.78 \approx 0.919c.\]

Conclusion

Each particle's speed in the laboratory frame is approximately \(0.919c\).

Key concepts

special relativity

Special relativity is a groundbreaking theory introduced by Albert Einstein in 1905. It revolutionized our understanding of time and space by showing how they are interconnected. The theory provides two key postulates:

• The laws of physics are the same in all inertial frames of reference.
• The speed of light in vacuum is constant and is independent of the motion of the observer.

These ideas lead to fascinating phenomena, such as time dilation and length contraction. When objects move close to the speed of light, these effects become significant. Consider high-speed particles in particle accelerators: their relativistic speeds require using special relativity to accurately describe their behavior. Without relativistic velocity addition, traditional physics would fail in calculating objects’ velocities accurately, especially when they reach near-light speeds.

quadratic equations

Quadratic equations are fundamental in mathematics. They are polynomial equations of the form:\[ ax^2 + bx + c = 0 \]where \(a\), \(b\), and \(c\) are constants. The solutions to these equations can be found using the quadratic formula:\[ x = \frac{-b \, \pm \, \sqrt{b^2 - 4ac}}{2a} \]Quadratic equations are often encountered in physics when processes have acceleration or squared terms, as seen in relativistic velocity calculations.
For example, in our relativistic velocity problem, the calculation simplifies into a quadratic form. Solving it determines the velocities of high-speed particles relative to the laboratory frame, crucial for experiments in physics labs and theoretical explorations alike. Understanding how to manipulate and solve quadratics becomes essential when encountering these types of real-world physics problems.

high-energy physics

High-energy physics is the field that explores the most fundamental components and forces in the universe. It investigates particles at extremely high energies, often at speeds close to that of light. The Large Hadron Collider (LHC) is an example of a facility that investigates particles such as protons at unprecedented speeds.
At these energetic extremes, traditional physics principles like Newtonian mechanics are inadequate. Instead, principles from quantum mechanics and special relativity are necessary to understand particle behaviors, such as collisions and creation of new particles. The research carried out in this field not only answers fundamental questions about the universe's origin but also leads to technological advancements. High-energy physics applications include improvements in computing, medicine (e.g., PET scans), and development of new materials.

particle accelerators

Particle accelerators are devices that propel charged particles, such as electrons or protons, to high speeds and energies. They allow scientists to collide these particles to study the fundamental laws of physics. By examining the resulting interactions, researchers can probe the constituents of matter and forces at a minute scale.

• Linear accelerators: Utilize electric fields to accelerate particles in a straight line.
• Synchrotrons: Use magnetic fields to accelerate particles in circular paths.

Accelerators are essential tools in both basic science and applied research. They are used not just in physics but also in fields like medicine, for radiation therapy, and materials science, for understanding the properties of substances at atomic levels.
The use of particle accelerators is heavily intertwined with high-energy physics, as they provide the necessary conditions to explore how particles behave at near light speeds, a regime where relativistic effects can't be ignored.