Question

Use light cones and world lines to help solve the following problem. Eddie and Martin are throwing water balloons very fast at a target. At \(t=-13 \mu s,\) the target is at \(x=0,\) Eddie is at \(x=-2 \mathrm{~km},\) and Martin is at \(x=5 \mathrm{~km},\) and all three remain in these positions for all time. The target is hit at \(t=0 .\) Who made the successful shot? Prove this using the light cone for the target. When the target is hit, it sends out a radio signal. When does Martin know the target has been hit? When does Eddie know the target has been hit? Use the world lines to show this. Before starting to draw your diagrams, consider: If your \(x\) position is measured in \(\mathrm{km}\) and you are plotting \(t\) versus \(x / c,\) what units must \(t\) be in, to the first significant figure?

Short answer

Answer: Eddie made the successful shot. Eddie knows the target has been hit at \(t_{receive-E} = 0\mu s + \frac{-2\mathrm{~km}}{c}\), and Martin knows the target has been hit at \(t_{receive-M} =0\mu s + \frac{5\mathrm{~km}}{c}\).

Step-by-step solution

Analyze the initial conditions

We are given the positions of Eddie, Martin, and the target at \(t=-13 \mu s\): - Eddie is at \(x = -2 \mathrm{~km}\) - Martin is at \(x = 5 \mathrm{~km}\) - Target is at \(x = 0\) All three remain in these positions for all time, and the target is hit at \(t=0\). First, let's address the unit question: If your \(x\) position is measured in \(\mathrm{km}\) and we are plotting \(t\) versus \(x/c\), we must convert \(t\) to units of \(\mu s\cdot c\) where \(c\) is the speed of light.

Determine the distance traveled by the radio signal

When the target is hit, it sends out a radio signal. Since radio signals travel at the speed of light (\(c\)), we need to find the distance each signal travels to reach Eddie and Martin. For Eddie, the distance is 2 km. For Martin, the distance is 5 km.

Determine when Eddie and Martin receive the signal

To find out when Eddie and Martin receive the signal, we will use the relationship of speed, distance, and time for each of them: - For Eddie, \(t_E = \frac{x_E}{c} = \frac{-2 \mathrm{~km}}{c}\) - For Martin, \(t_M = \frac{x_M}{c} = \frac{5 \mathrm{~km}}{c}\)

Analyze the light cone for the target

To determine who hit the target, we'll analyze the light cone for the target at \(t=0\). Since the target is at the origin, we consider the past light cone (the area within the light cone at earlier times). If Eddie or Martin's world line intersects the past light cone at \(t=-13\mu s\), that person made the successful shot.

Determine the intersection point of the world lines with the past light cone of the target

Equate the time it takes for a signal to reach Eddie with the time it takes for a signal to reach Martin: - \(t_E=-13\mu s+\frac{x_E}{c}\) - \(t_M=-13\mu s+\frac{x_M}{c}\) If the intersection points for Eddie and Martin's world lines are in the light cone at earlier times, we know who made the shot. Evaluating the intersection times: - \(t_E = -13\mu s + \frac{-2\mathrm{~km}}{c} > -13\mu s\) which lies inside the past light cone. - \(t_M = -13\mu s + \frac{5\mathrm{~km}}{c} > -13\mu s\) which lies outside the past light cone.

Determine the successful shot and when they know the target has been hit

Since Eddie's world line intersects the past light cone and Martin's world line doesn't, Eddie made the successful shot. To find out when Eddie and Martin know the target has been hit, we calculate the time it takes for the radio signal to reach them using the time values from Step 3: - For Eddie, \(t_{receive-E} = 0\mu s + \frac{-2\mathrm{~km}}{c}\) - For Martin, \(t_{receive-M} = 0\mu s + \frac{5\mathrm{~km}}{c}\)

Key concepts

Special Relativity

Special relativity is a fundamental theory in physics introduced by Albert Einstein in 1905, which transformed our understanding of space and time. It is based primarily on two postulates: the laws of physics are the same in all inertial frames of reference, and the speed of light in vacuum is constant and independent of the observer's motion. The ramifications of these postulates include time dilation and length contraction, leading to the conclusion that events simultaneous in one frame of reference may not be in another.

Within the context of our water balloon exercise, special relativity would imply that both Eddie and Martin are in different frames of reference relative to the target, and thus the concepts of simultaneity and the invariance of the speed of light are key to solving the problem. When we're dealing with events that involve high speeds or significant distances, such as those involving light or radio signals, we must apply the principles of special relativity to determine the sequence of events accurately. This approach ensures the clarity of the solution and helps to grasp the essence of relativity in the context of light signals and their perception by different observers.

Speed of Light

The speed of light, denoted by the symbol 'c', is a constant that holds a fundamental place in the field of physics, especially in the theory of special relativity. The exact value of the speed of light in a vacuum is approximately 299,792 kilometers per second. One of the profound corollaries of this constancy is that it serves as a cosmic speed limit: no object or signal can travel faster than the speed of light.

Regarding the exercise on world lines and light cones, this constant 'c' plays a crucial role in calculating the time it takes for a radio signal (which travels at the speed of light) to reach Eddie and Martin from the target. As we saw in the exercise steps, the use of 'c' directly impacts the time calculations and the understanding of who hit the target and when the participants are aware of the impact. By ensuring we maintain consistency by using units that respect the speed of light, students can better appreciate the nuances of light's role in special relativity and the outcomes of the exercise.

Space-Time Diagrams

Space-time diagrams are visual tools used in the realm of special relativity to represent the events in a two-dimensional plane, where one axis is dedicated to space (for instance, distance 'x'), and the other to time ('t'). Light cones and world lines are critical components of these diagrams. A light cone represents the path a flash of light would take through space-time; its structure demonstrates that the speed of light serves as a boundary which cannot be surpassed by any physical object or non-light signal.

In our example, the exercise invites students to determine who hit the target and when the participants learned about the hit. By plotting the world lines, which track the movement (or, as in the exercise, the stationary position) of Eddie and Martin against the light cone originating from the event of the target being hit, one can visually ascertain which participant made the successful shot. World lines that intersect with the light cone's past reflect possible paths from which the target could have been reached. Students who can effectively utilize space-time diagrams are better equipped to comprehend the conceptual underpinnings of events in special relativity. The use of clear, simple diagrams, accompanied by straightforward calculations, is vital for students to accurately map out and understand the sequence of events within this relativistic framework.