Problem 46 Observers in reference frame \(S... [FREE SOLUTION] (2024)

Chapter 1: Problem 46

Observers in reference frame \(S\) see an explosion located at \(x_{1}=480\mathrm{~m}\). A second explosion occurs \(5 \mu\) s later at \(x_{2}=1200\mathrm{~m}\). In reference frame \(S^{\prime}\), which is moving along the \(+x\)axis at speed \(v\), the explosions occur at the same point in space. ( \(a\) )Draw a spacetime diagram describing this situation. ( \(b\) ) Determine \(v\) fromthe diagram. ( \(c\) ) Calibrate the \(c t^{\prime}\) axis and determine theseparation in time in \(\mu\) s between the two explosions as measured in\(S^{\prime} .(d)\) Verify your results by calculation.

Short Answer

Expert verified

The velocity v can be calculated using Lorentz transformation: set x1' = x2', with x' = gamma(x - vt) and gamma = 1/sqrt(1-v^2/c^2), then solve for v. To find the time separation in S', use the time transformation equation t' = gamma(t - vx/c^2). Verify both velocity and time by plugging the values back into the Lorentz transformation equations.

Step by step solution

Understanding the Problem

We are given two events characterized by their positions and times in reference frame S. We need to find the velocity v of another reference frame S' in which both events occur at the same spatial point. First step is to understand the data given and how spacetime diagrams represent events in different reference frames.

Drawing the Spacetime Diagram for Frame S

On the spacetime diagram, plot frame S's x-axis horizontally and ct-axis vertically (where c is the speed of light). Mark the points for the two explosions, with one at (480 m, 0) and the other at (1200 m, 5 microseconds). The ct-axis will remain unchanged since it is the proper time axis for frame S.

Identifying the Slope for Frame S'

We know that in frame S', both events must occur at the same spatial location, thus they lie along a line parallel to the ct'-axis. The slope of this line corresponds to the velocity v of the S' frame, and it will cross the two events when plotted correctly.

Determining Velocity v from the Diagram

The slope of the S' frame's ct'-axis (which is the trajectory of the S' origin) in the diagram can be determined using the rise over run between the two events. The slope should be delta(ct)/delta(x), and since we want velocity v, we take its inverse, which gives us delta(x)/delta(ct), where delta(x) is the spatial and delta(ct) is the temporal separation of the two events in frame S.

Calibrating the ct' axis

To calibrate the ct' axis, we need to consider the time dilation effect. The ct' axis will be at an angle to the ct axis, corresponding to the relative velocity v. Since the events are simultaneous in S', the ct' coordinates for both events will be the same, so the separation in ct' is zero.

Calculating Time Separation in S'

We can use the Lorentz transformation to find the time separation in reference frame S'. The events are at the same location in S', so delta(x') = 0. Using the Lorentz transformation, we calculate delta(t') knowing delta(x), delta(t), and velocity v.

Quantitative Calculation of v

The Lorentz transformation for spatial coordinates gives us: x' = gamma(x - vt), where gamma = 1/sqrt(1-(v^2/c^2)). Since both explosions are observed at the same point in S', set x'_1 = x'_2, plug in the values for x_1, x_2, and delta t = 5 microseconds, and solve for v.

Verification by Calculation

To verify, use the determined value of v to check if it satisfies the Lorentz transformation conditions. Substituting back into the equations to ensure that x'_1 = x'_2 should yield true, confirming that the velocity v is correct and the time separation in S' is consistent with Lorentz transformation equations.

Key Concepts

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

Spacetime Diagram

Visualizing the concept of events and how they occur across different spatial positions and times can be quite challenging without a proper tool; enter the spacetime diagram. A spacetime diagram is a two-dimensional graphical representation where time and space coordinates are plotted to illustrate the relationship between different events in the universe. In the spacetime diagram, the horizontal axis typically represents space, while the vertical axis represents time, often scaled by the speed of light (denoted by c) for relativistic purposes.

The power of a spacetime diagram lies in its ability to show how different observers perceive events. For instance, in the given exercise, we have two explosions separated by distance and time. Plotting these on a spacetime diagram, we can visualize their separations and see the effects of moving reference frames. Events that are simultaneous in one frame will not necessarily be simultaneous in another, which is precisely what the exercise demonstrates.

Lorentz Transformation

To connect the dots between different observers' perceptions, we can't rely on intuition alone; we need the mathematics of the Lorentz transformation. These equations are the backbone of special relativity, allowing us to calculate how measurements of time and space taken by one observer relate to those taken by another, moving at a constant velocity relative to the first.

The Lorentz transformation takes into account the finiteness of the speed of light and shows that time and space are not absolute but relative and intertwined—hence the term 'spacetime.' When solving our textbook problem, we use the Lorentz transformation to find the velocity of the moving frame S' that renders the two explosions simultaneous, and later to calculate the time separation between the same events as seen from S'.

Time Dilation

Among the counterintuitive concepts that special relativity presents, time dilation is a prominent one. It tells us that time doesn't tick at the same rate for everyone; it's affected by the relative velocity between observers. The faster you move compared to someone else, the slower your clock ticks from their perspective.

When calibrating the ct' axis in a spacetime diagram for a moving reference frame, we must consider time dilation. In our exercise, even though the explosions happen 5 microseconds apart in frame S, they are not necessarily the same amount of time apart in frame S'. This discrepancy needs to be determined using the relative velocity found through the Lorentz transformation, showcasing time dilation in action.

Reference Frames

Another cornerstone idea of special relativity is that of reference frames. A reference frame is simply a vantage point from which an observer makes measurements in space and time. Reference frames can be stationary or in motion relative to each other, and moving frames exhibit the fascinating effects predicted by relativity, such as time dilation and length contraction.

In the given problem, we look at two reference frames: S and S'. The frame S is stationary, while S' moves at a certain velocity. Understanding how different reference frames perceive the same events is essential to solving problems in special relativity. Through the exercise, we determine the velocity at which frame S' must move so that two spatially separated events appear simultaneous, highlighting the relativity of simultaneity in different frames.

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