Problem 92 - Fundamentals of Physics

Question

A relativistic train of proper length \(200 \mathrm{~m}\) approaches tunnel of the same proper length, at a relative speed of \(0.900 c\). A paint bomb in the engine room is set to explode (and cover everyone with blue paint) when the front of the train passes the far end of the tunnel (event FF). However, when the rear car passes the near end of the tunnel (event \(\mathrm{RN}\) ), a device in that car is set to send a signal to the engine room to deactivate the bomb. Train view: (a) What is the tunnel length? (b) Which event occurs first, FF or RN? (c) What is the time between those events? (d) Does the paint bomb explode? Tunnel view: (e) What is the train length? (f) Which event occurs first? (g) What is the time between those events? (h) Does the paint bomb explode? If your answers to (d) and (h) differ, you need to explain the paradox, because either the engine room is covered with blue paint or not; you cannot have it both ways. If your answers are the same, you need to explain why.

Step-by-Step Solution

Answer

In the train's view, the bomb doesn't explode, but in the tunnel's view, it does. This apparent paradox is explained by relativity of simultaneity.

Step 1: Determine the Length of the Tunnel (Train View)

In the train's reference frame, the tunnel is moving at a speed of 0.900c, so we need to calculate the contracted length of the tunnel. Using the length contraction formula:\[ L = L_0 \sqrt{1 - v^2/c^2} \]where \(L_0 = 200 \text{ m}\) is the proper length of the tunnel, and \(v = 0.900c\):\[ L = 200 \text{ m} \times \sqrt{1 - (0.900)^2} = 200 \times \sqrt{1 - 0.81} = 200 \times \sqrt{0.19} \approx 87 \text{ m}\]So, the length of the tunnel in the train's frame is approximately 87 meters.

Step 2: Determine Event Order (Train View)

In the train's reference frame, the contracted tunnel length is shorter than the train's proper length. Therefore, it observes that the front of the train exits the tunnel before the rear enters it, thus RN occurs before FF in the train view.

Step 3: Calculate Time Between Events (Train View)

The time between events FF and RN can be determined by considering the relative speed of the train and tunnel and the distance (i.e., the contracted length of the tunnel). Using\[ \Delta t = \frac{L}{v} = \frac{87 \text{ m}}{0.900c} \approx \frac{87}{0.900} \times \frac{1}{c} \approx 0.322 \times \frac{1}{c} \]Converting \(c\) (speed of light) into meter per second (\(c \approx 3 \times 10^8 \text{ m/s}\)) yields a time interval:\[ \Delta t \approx 0.322 \times \frac{1}{3 \times 10^8} \approx 1.07 \times 10^{-9} \text{ s} \]So, the time gap between RN and FF event from the train's view is about \(1.07 \text{ nanoseconds}\).

Step 4: Determine Explosion Event (Train View)

Since RN (signal from the rear) occurs before FF (front of train reaching the end of the tunnel) in the train's view, the signal to deactivate the bomb reaches the engine room before the bomb detonates. Therefore, the paint bomb does not explode in the train's frame.

Step 5: Calculate Train Length (Tunnel View)

In the tunnel's reference frame, it's the train that is contracted. Using length contraction for the train moving at 0.900c:\[ L = L_0 \sqrt{1 - v^2/c^2} \]where \(L_0 = 200 \text{ m}\):\[ L = 200 \text{ m} \times \sqrt{1 - (0.900)^2} = 200 \times \sqrt{0.19} \approx 87 \text{ m}\]Thus, the contracted length of the train from the tunnel's view is also about 87 meters.

Step 6: Determine Event Order (Tunnel View)

In the tunnel's frame, where the contracted train length equals tunnel length, both events RN and FF occur simultaneously because the contracted train just fits the tunnel. Therefore, there is no delay between RN and FF.

Step 7: Calculate Time Between Events (Tunnel View)

Since in the tunnel's frame both RN and FF occur simultaneously, the time interval between these events is zero.

Step 8: Determine Explosion Event (Tunnel View)

As both events occur simultaneously in the tunnel's view, the signal to deactivate the bomb does not reach before the bomb’s trigger, leading to the conclusion that the paint bomb explodes.

Step 9: Explain the Paradox

The difference in result (a paradox) between the frames is resolved by recognizing the relativity of simultaneity. The order of events is different because time is not absolute; it changes based on the observer's relative motion. Thus, both views are correct within their frames of reference, but in practice, consistent rules like signal priority in engineering setups determine one practical outcome — favoring the design drafters. Without an explicit default in this problem description, assuming physics consistency as material frameworks (designed switch precedence) will hold in our real universe.

Key Concepts

Length Contraction

In the fascinating world of relativity, length contraction is a core concept. It describes how objects appear shorter in the direction of motion when observed from a different reference frame that is moving relative to the object. When we talk about "proper length," we refer to the length of an object as measured in the object's own rest frame - in this exercise, both the train and the tunnel have this proper length of 200 meters.
Now, let's look at the train's point of view. As the train moves at 0.900c relative to the tunnel, it sees the tunnel as shorter than its proper length due to length contraction. The formula to calculate this is:\[L = L_0 \sqrt{1 - \left(\frac{v}{c}\right)^2}\]where \(L_0\) is the proper length and \(v\) is the relative speed. Substituting the given values, the tunnel appears to be about 87 meters to the train. This contraction is critical, as it affects how the train perceives whether it fully fits in the tunnel at any given instance.

Relativity of Simultaneity

The relativity of simultaneity is one of Einstein's revelations, which states that two events happening simultaneously in one reference frame may not be simultaneous in another. This is pivotal in understanding the seeming paradox in the train and tunnel exercise.
From the train’s view, because the tunnel appears shorter than the train, it sees the rear passing the near end (RN) of the tunnel first. The signal is sent before the front exits the tunnel (FF), allowing the bomb to be deactivated in time. However, from the tunnel's perspective, both events (RN and FF) happen at the same time due to its view of the contracted train fitting exactly. This results in the bomb exploding as there is no time for the signal to communicate.
Thus, relativity of simultaneity teaches us that which event occurs first isn't absolute, but depends entirely on the observer’s frame of motion.

Proper Length

Proper length is the term we use to denote the length of an object in its own rest frame. For the train and the tunnel in our problem, their proper lengths are both measured as 200 meters when stationary relative to themselves.
Proper length is crucial to understand because it's a measure unaffected by motion. When either the train or the tunnel is observed from another reference frame moving relative to it, this length will appear contracted. Therefore, it helps frame exactly how much contraction will be perceived by any observer not stationary relative to the object, using the length contraction formula. Proper length allows us to calculate how much of that contraction is purely due to relative motion, a key aspect of understanding scenarios in relativity.

Reference Frame

The concept of a reference frame is fundamental to understanding not just relativity, but all physics. It is essentially a point of view or coordinate system through which observers measure and perceive phenomena.
In this exercise, we primarily deal with two reference frames: that of the train and that of the tunnel. Each has its own perception of events, lengths, and time intervals. The train's reference frame sees its own length as unaffected by motion while the tunnel is contracted; conversely, the tunnel’s frame sees itself at its proper length and the train contracted.
Analyzing the puzzle of the paint bomb through different reference frames reveals how observers can experience different realities of the same event according to their relative motion. Understanding varying observations across these frames without perceiving contradictions is one of the ways relativity enriches our grasp of the universe.