Problem 16 - Fundamentals of Physics

Question

Observer \(S\) reports that an event occurred on the \(x\) axis of his reference frame at \(x=3.00 \times 10^{8} \mathrm {~m}\) at time \(t=2.50 \mathrm{~s}\). Observer \(S^{\prime}\) and her frame are moving in the positive dircction of the \(x\) axis at a speed of \(0.400 \mathrm{c}\). Further, \(x=x^{\prime}=0\) at \(t=t^{\prime} =0\), What are the (a) spatial and (b) temporal coordinate of the cvent according to \(S^{\prime \prime}\) If \(S^{\prime}\) were, instead, moving in the negative direction of the \(x\) axis, what would be the (c) spatial and (d) temporal coordinate of the cvent according to \(S^ {\prime \prime}\) ?

Step-by-Step Solution

Answer

For observer S', when moving positively: (a) x' ≈ 2.84×10^8 m, (b) t' ≈ 2.07 s. Moving negatively: (c) x' ≈ 3.16×10^8 m, (d) t' ≈ 2.93 s.

Step 1: Identify Given Variables for First Scenario

For observer \( S \), we have the parameters \( x = 3.00 \times 10^8 \text{ m} \) and \( t = 2.50 \text{ s} \). For observer \( S' \), who moves in the positive \( x \)-direction at \( v = 0.400c \), we need to find the new coordinates \( x' \) and \( t' \).

Step 2: Calculate Lorentz Factor

The Lorentz factor \( \gamma \) is needed for the transformation equations and is given by:\[ \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} \]Substitute \( v = 0.400c \) to find:\[ \gamma = \frac{1}{\sqrt{1 - (0.400)^2}} \approx 1.091 \]

Step 3: Perform Lorentz Transformation for Space Coordinate in Positive Direction

The transformation for the space coordinate is given by:\[ x' = \gamma (x - vt) \]Substitute the known values:\[ x' = 1.091 \left(3.00 \times 10^8 \text{ m} - 0.400c \times 2.50 \text{ s}\right) \]Caring out this calculation:\[ x' = 1.091 \left(3.00 \times 10^8 \text{ m} - 0.400 \times 2.50 \times 3.00 \times 10^8 \right) \approx 2.84 \times 10^8 \text{ m} \]

Step 4: Perform Lorentz Transformation for Time Coordinate in Positive Direction

The transformation for the time coordinate is:\[ t' = \gamma \left( t - \frac{vx}{c^2} \right) \]Substitute the known values:\[ t' = 1.091 \left(2.50 \text{ s} - \frac{0.400 \times 3.00 \times 10^8 \text{ m}}{3.00 \times 10^8 \text{ m/s}}\right) \]Calculating this:\[ t' = 1.091 \times (2.50 - 0.400)( \text{s} ) \approx 2.07 \text{ s} \]

Step 5: Repeat for Observer Moving in Negative Direction

Now assume \( S' \) is moving in the negative direction with \( v = -0.400c \). Recalculate spatial coordinate:\[ x' = \gamma (x - vt) = 1.091 \left(3.00 \times 10^8 \text{ m} + 0.400 \times 2.50 \times 3.00 \times 10^8 \right) \]Calculating:\[ x' \approx 3.16 \times 10^8 \text{ m} \]

Step 6: Perform Temporal Coordinate Transformation in Negative Direction

For time, using \( v = -0.400c \):\[ t' = \gamma \left( t + \frac{vx}{c^2} \right) = 1.091 \left(2.50 + 0.400 \right) \text{ s} \approx 2.93 \text{ s} \]

Key Concepts

Special Relativity

Special relativity is a groundbreaking theory proposed by Albert Einstein in 1905. This theory revolutionized our understanding of space and time, asserting that the laws of physics are the same for all non-accelerating observers. One of the core principles of special relativity is the constancy of the speed of light. The speed of light in a vacuum is the same for all observers, regardless of their relative motion.
This reconciled issues seen in classical mechanics with electromagnetism.

• Time Dilation: Time can dilate, or lengthen, depending on the relative velocity between observers. As an object moves closer to the speed of light, time for it slows down relative to a stationary observer.
• Length Contraction: Objects moving at speeds close to the speed of light will appear shorter in the direction of motion when observed from a stationary frame.

These phenomena, although counterintuitive, are crucial in describing the behavior of particles moving near light speed. Together, they form the basis of many calculations, including the Lorentz transformation used in our exercise.

Relativistic Physics

Relativistic physics takes into account the principles of relativity, considering objects moving at velocities close to the speed of light. Unlike classical physics, which worked well for everyday speeds, relativistic physics explains phenomena at velocities where relativistic effects become significant.
It introduces concepts like:

• Mass-Energy Equivalence: Energy and mass are interchangeable, as represented by the famous equation, \(E=mc^2\).
• Relativistic Momentum: Momentum increases significantly as an object's velocity approaches the speed of light.

These concepts help understand the interactions of objects in high-speed contexts, such as particle physics and astrophysics. Lorentz transformation equations are a crucial tool in this field, allowing us to calculate coordinates in different reference frames under relativistic conditions.

Reference Frames

In physics, a reference frame is a perspective or set of conditions from which observations are made and measurements are taken. In special relativity, it's essential to define reference frames since motion is relative.
Here are some key points:

• Inertial Reference Frame: A frame where objects are either at rest or move at constant velocity (no acceleration). It is the framework of special relativity.
• Relativity of Simultaneity: Events that are simultaneous in one reference frame may not be simultaneous in another. This implies that space and time are interconnected.

The exercise involves transforming coordinates from one reference frame, S, to another, S', using the Lorentz transformation.
This procedure ensures accurate measurement of spatial and temporal coordinates when the observer and object are in relative motion at high speeds.