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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

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

Step by step solution

01

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.
02

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:\[0.890c = \frac{2v}{1 + \frac{v^2}{c^2}}.\]
03

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:\[0.890c + 0.890 \frac{v^2}{c} = 2v.\]
04

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 \(a = 0.890, b = -2c, c = 0.890c^2\):\[v = \frac{2c \pm \sqrt{(-2c)^2 - 4(0.890)(0.890c^2)}}{2(0.890)}.\]
05

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:\[v = \frac{2c + 0.9119c}{1.78} = \frac{2.9119c}{1.78} = 1.6353c/1.78 \approx 0.919c.\]
06

Conclusion

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

A spaceship flies past Mars with a speed of 0.985c relative to the surface of the planet. When the spaceship is directly overhead, a signal light on the Martian surface blinks on and then off. An observer on Mars measures that the signal light was on for 75.0 ms. (a) Does the observer on Mars or the pilot on the spaceship measure the proper time? (b) What is the duration of the light pulse measured by the pilot of the spaceship?

Everyday Time Dilation. Two atomic clocks are carefully synchronized. One remains in New York, and the other is loaded on an airliner that travels at an average speed of 250 m/s and then returns to New York. When the plane returns, the elapsed time on the clock that stayed behind is 4.00 h. By how much will the readings of the two clocks differ, and which clock will show the shorter elapsed time? (\(Hint\): Since \(u \ll c\), you can simplify \(\sqrt{1 - u^2/c^2}\) by a binomial expansion.)

The positive muon (\(\mu^+\)), an unstable particle, lives on average 2.20 \(\times\) 10\(^{-6}\) s (measured in its own frame of reference) before decaying. (a) If such a particle is moving, with respect to the laboratory, with a speed of 0.900c, what average lifetime is measured in the laboratory? (b) What average distance, measured in the laboratory, does the particle move before decaying?

A meter stick moves past you at great speed. Its motion relative to you is parallel to its long axis. If you measure the length of the moving meter stick to be 1.00 ft 11 ft = 0.3048 m2-for example, by comparing it to a 1-foot ruler that is at rest relative to you-at what speed is the meter stick moving relative to you?

An alien spacecraft is flying overhead at a great distance as you stand in your backyard. You see its searchlight blink on for 0.150 s. The first officer on the spacecraft measures that the searchlight is on for 12.0 ms. (a) Which of these two measured times is the proper time? (b) What is the speed of the spacecraft relative to the earth, expressed as a fraction of the speed of light \(c\)?

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