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Chapter 38: Problem 33

A particle moves with uniform speed \(v_{y}^{\prime}=\Delta y^{\prime} / \Delta t^{\prime}\) in the \(y^{\prime}\) direction with respect to a rocket frame that moves along the \(x\) axis of a laboratory frame. Find exprcssions for the \(x\) -componcnt and for the \(y\) -componcnt of the particle's velocity in the laboratory frame.

Short Answer

Expert verified
The x-component is \( v_x = u_x \) and the y-component is \( v_y = v_y' \sqrt{ 1- \frac{u_x^2}{c^2} } \).

Step by step solution

01

Identify the given values and variables

A particle moves with uniform speed in the y' direction with respect to a rocket frame. Speed in the rocket frame: \( v_{y}^{\text{'} }= \frac{\Delta y^{ \text{'} }} {\Delta t^{ \text{'} }} \), and the rocket is moving along the x axis of a laboratory frame.
02

Understand the relationship between frames

The rocket frame moves with velocity \( v_x \) along the x-axis with respect to the laboratory frame. Let \( V \) be the velocity of the particle relative to the laboratory frame.
03

Apply the Lorentz Transformation for velocities along the x-axis

For the x-axis, the velocity transformation is given by: \[ v_x = \frac{ u_x + v_x' }{ 1 + \frac{ u_x v_x' }{c^2} } \] where \( u_x \) is the velocity of the rocket along the x-axis.
04

Determine the x-component of the particle's velocity in the lab frame

Since the rocket is moving in the x direction and the particle in the y' direction, \( v_x' = 0 \). Therefore, the x-component of the particle's velocity in the laboratory frame is: \[ v_x = u_x \]
05

Apply the Lorentz Transformation for velocities along the y-axis

For the y-axis, the velocity transformation is given by: \[ v_y = \frac{ v_y' }{ \gamma ( 1 + \frac{ u_x v_x' }{c^2} ) } \] where \( \gamma \) is the Lorentz factor: \( \gamma = \frac{1}{\sqrt{1-\frac{u_x^2}{c^2}}} \). With \( v_x' = 0 \), it simplifies to: \[ v_y = \frac{ v_y' }{ \gamma } \]
06

Obtain the y-component of the particle's velocity in the lab frame

Using the Lorentz factor \( \gamma = \frac{1}{\sqrt{1-\frac{u_x^2}{c^2}}} \), the y-component of the particle's velocity in the laboratory frame is: \[ v_y = v_y' \sqrt{ 1- \frac{u_x^2}{c^2} } \]

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Key Concepts

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

Relative Motion
Relative motion deals with the movement of objects as observed from different reference frames. Imagine you're on a train. To you, you might be sitting still, but to someone standing on a platform, you are moving. This is what physicists mean by 'relative motion.' When dealing with relative motion, it is important to clearly define from which point of view you are observing the movement. Often in physics, we look at motion from a 'lab frame' (a stationary observer) and a 'moving frame' (like a person on the train). Understanding these different perspectives helps us better understand how objects move in different situations.
Velocity Transformation
Velocity transformation is essential when dealing with objects observed from different reference frames, especially in the context of special relativity. When an object moves, its velocity can be observed differently by different observers. Special relativity introduces Lorentz transformations, which help in adjusting velocities seen from one frame to another. For example, if a particle is moving in a rocket (moving frame) in the y direction, we need to transform this to the lab frame using specific formulas. For the x component, where the particle moves in the x-axis, the velocity transformation simplifies if the initial velocity in that direction is zero. In our case, the x-component of velocity remains the velocity of the rocket itself. For the y component, the transformation considers the Lorentz factor. This correctly translates the particle's speed in the y direction from the moving to the stationary frame.
Special Relativity
Special relativity, proposed by Albert Einstein, revolutionized our understanding of space and time. One of its key ideas is that the laws of physics are the same for all non-accelerating observers, and the speed of light is constant irrespective of the observer's state of motion. Special relativity introduces the concept of time dilation and length contraction, explaining why time appears to slow down and objects contract along the direction of motion as they approach the speed of light. A fundamental aspect of special relativity is the Lorentz transformation, which includes the famous Lorentz factor \(\gamma\). This transformation adjusts measurements like time and velocity between different reference frames. For our exercise, it is used to transform the particle's velocities from the rocket frame to the lab frame, ensuring accurate descriptions of motion across different perspectives.

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Most popular questions from this chapter

The values of the masses in the reaction \(p+{ }^{19} F \rightarrow \alpha+{ }^{16} O\) have been determined by a mass spectrometer to have the values: $$ \begin{aligned} m(p) &=1.007825 u, \\ m(F) &=18.998405 u, \\ m(\alpha) &=4.002603 u, \\ m(O) &=15.994915 u . \end{aligned} $$ Here \(u\) is the atomic mass unit (Section 1.7). How much energy is released in this reaction? Express your answer in both kilograms and \(\mathrm{MeV}\).

A spaceship moving away from Earth at a speed \(0.900 c\) radios its reports back to Earth using a frequency of 100 MHz measured in the spaceship frame. To what frequency must Earth's receivers be tuned in order to receive the reports?

A spaceship of rest length \(100 \mathrm{~m}\) passes a laboratory timing station in \(0.2\) microseconds measured on the timing station clock. (a) What is the speed of the spaceship in the laboratory frame? (b) What is the Lorentz-contracted length of the spaceship in the laboratory frame?

(a) Can a person, in principle, travel from Earth to the center of our galaxy, which is 23000 ly distant, in one lifetime? Explain using either length contraction or time dilation arguments. (b) What constant speed with respect to the galaxy is required to make the trip in \(30 \mathrm{y}\) of the traveler's lifetime?

You and a group of female and male friends stand outdoors at dusk watching the Sun set and noticing the planet Venus in the same direction as the Sun. An alien ship lands beside you at the same instant that you see the Sun explode. The aliens admit that earlier they shot a laser flash at the Sun, which caused the explosion. They warn that the Sun's explosion emitted an immense pulse of particles that will blow away Earth's atmosphere. In confirmation, a short time after the aliens land you notice Venus suddenly change color. You and your friends plead with the aliens to take your group away from Earth in order to establish the human gene pool elsewhere. They agree. Describe the conditions under which your escape plan will succeed. Be specific and use numbers. Assume that the Sun is 8 light-minutes from Earth and Venus is 2 light-minutes from Earth.
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