Problem 18 - Fundamentals of Physics

Question

Inertial frame \(S^{\prime}\) moves at a speed of \(0.60 c\) with respect to frame \(S\) (Fig. 37-9). Further, \(x=x^{\prime}=0\) at \(t=t^{\prime}=0\). Two events are recorded. In frame \(S\), event 1 occurs at the origin at \(t=0\) and event 2 occurs on the \(x\) axis at \(x=3.0 \mathrm{~km}\) at \(t=4.0 \mu \mathrm{s}\). According to observer \(S^{\prime}\), what is the time of (a) event 1 and (b) event \(2 ?\) (c) Do the two observers see the same sequence or the reverse?

Step-by-Step Solution

Answer

(a) Event 1 is at \( t' = 0 \) in \( S' \), (b) Event 2 at \( t' \approx -7.5 \) ms, (c) Sequence is reversed.

Step 1: Understand the Problem

We have two inertial frames, \( S \) and \( S' \), and we are given information about events in frame \( S \). The task is to determine the timing of the events in frame \( S' \). The velocity of \( S' \) with respect to \( S \) is \( 0.60 c \). We need to find out when events occur in \( S' \) and determine the sequence of events.

Step 2: Use Lorentz Transformation Equations

Lorentz transformation equations relate the coordinates of events in two inertial frames. For time, the transformation is: \[ t' = \gamma (t - \frac{vx}{c^2}) \]where \( v = 0.60c \), \( \gamma = \frac{1}{\sqrt{1-v^2/c^2}} \), \( x \) is the position of the event, and \( t \) is the time in frame \( S \). Calculate \( \gamma \) first:\[ \gamma = \frac{1}{\sqrt{1-(0.60)^2}} = \frac{1}{0.8} = 1.25 \]

Step 3: Determine Time of Event 1 in Frame S'

For event 1, \( x=0 \) and \( t=0 \) in frame \( S \). Using the transformation:\[ t' = \gamma (t - \frac{vx}{c^2}) = 1.25(0 - 0) = 0 \] Therefore, event 1 occurs at \( t' = 0 \) in frame \( S' \).

Step 4: Determine Time of Event 2 in Frame S'

For event 2, substitute \( x = 3000 \, \text{m} \) (since \( 3.0 \, \text{km} = 3000 \, \text{m} \)) and \( t = 4.0 \, \mu \text{s} \) in the transformation:\[ t' = 1.25 (4.0 \times 10^{-6} - \frac{(0.60 \times 3 \times 10^8)(3000)}{(3 \times 10^8)^2}) \]Simplifying:\[ t' = 1.25 (4.0 \times 10^{-6} - (0.60)(0.01)) \]\[ t' = 1.25 (4.0 \times 10^{-6} - 6.0 \times 10^{-3}) = 1.25 (-5.996 \times 10^{-3}) = -7.495 \times 10^{-3} \, \text{s} \]Therefore, event 2 occurs at \( t' \approx -7.5 \, \text{ms} \).

Step 5: Analyze the Sequence of Events

Event 1 occurs at \( t' = 0 \) and event 2 occurs at approximately \( t' = -7.5 \, \text{ms} \) in frame \( S' \). Since event 2 has a negative time relative to event 1, it happens before event 1 in the frame \( S' \). Thus, the sequence is reversed in \( S' \) compared to \( S \).

Key Concepts

Inertial Frames

In physics, an inertial frame is a reference frame where objects stay at rest or in uniform motion unless an external force acts upon them. This concept is crucial in understanding how observations differ for observers in different states of motion. Consider two inertial frames: frame \( S \) and frame \( S' \). These frames are moving relative to each other at a constant velocity. In the given exercise, frame \( S' \) moves at a velocity of \( 0.60c \) relative to frame \( S \), making them distinct inertial frames.

When we analyze events (like the recorded events in the exercise), each observer in different inertial frames may measure different spatial and temporal coordinates due to their relative motion. This is the basis of Einstein's theory of relativity, which states no inertial frame is preferred, and physical laws hold true in all inertial frames.

Understanding inertial frames helps us realize that what appears simultaneous or consecutive in one frame may not be the same in another, highlighting the relative nature of time and space.

Time Dilation

Time dilation is one of the intriguing concepts from Einstein's theory of relativity. It reflects how time can elapse at different rates for observers based on their relative velocity. According to the Lorentz transformation equations, the time coordinate in one inertial frame can differ from that in another due to their respective motion.

In the problem, observers in frames \( S \) and \( S' \) measure different times for the same events because of their high relative velocity \( 0.60c \). The mathematical expression for time dilation is represented by the equation:

• \( t' = \gamma(t - \frac{vx}{c^2}) \)

where \( \gamma = \frac{1}{\sqrt{1-v^2/c^2}} \) is the Lorentz factor. Using this factor, we can calculate the time experienced between events for an observer in another frame.

Through this exercise, we see event 1 occurs at \( t' = 0 \), while event 2 takes place at \( t' \approx -7.5 \text{ ms} \) in frame \( S' \), indicating a time shift caused by their relative motion. Time dilation shows that the fast-moving observer perceives time differently, often experiencing events in a different order or duration compared to a stationary observer. This leads us to the understanding that time is not absolute but rather a variable quantity that can change based on one's velocity.

Relativity of Simultaneity

The relativity of simultaneity is a profound implication of special relativity. It states that simultaneous events in one inertial frame might not be simultaneous in another. This principle becomes evident when events are observed from different frames moving relative to each other.

In the exercise provided, we see this concept in action. For observer \( S \), event 1 happens at the origin \((x=0)\) at \(t=0\) and event 2 occurs at \(x=3.0 \text{ km}\) at \(t=4.0 \mu\text{s}\). However, when looking at these events from frame \( S' \), the time calculated for event 2 (t' \approx -7.5 \text{ ms}) precedes that for event 1 \(t' = 0\). This calculates sequence reversal when viewed from \( S' \).

This demonstrates that the simultaneity of events is relative to the observer's frame of reference, challenging the notion of a universal present. Different observers can have their unique reality, seeing events in different orders or even perceiving overlapping instances as occurring at separate times. The relativity of simultaneity emphasizes that our experience of time and order is deeply connected to our state of motion.