Problem 92 - Fundamentals of Physics

Question

A relativistic train of proper length \(200 \mathrm{~m}\) approaches a 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

The paint bomb does not explode in both frames.

Step 1: Calculate Tunnel Length in Train Frame

According to the theory of relativity, lengths contract in the direction of motion as perceived from a different inertial frame. The tunnel's proper length is given as \( L_0 = 200 \) m. The tunnel length in the train's frame can be calculated using the length contraction formula:\[ L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \]Substitute \( v = 0.900c \):\[ L = 200 \sqrt{1 - (0.900)^2} = 200 \times \sqrt{0.19} \approx 86.6 \text{ m} \].

Step 2: Determine Which Event Occurs First in Train Frame

By analyzing the sequence of events from the train's perspective: the train perceives the tunnel to be only 86.6 m long. Since the train's length remains its proper length of 200 m in its own frame, the rear end of the train will pass the near end (event RN) before the front of the train passes the far end (event FF) because the whole train cannot fit within this shortened tunnel.

Step 3: Calculate Time Between Events in Train Frame

The events FF and RN are spatially separated events in the train frame. The time it takes for the rear of the train to cover the tunnel length of 86.6 m is calculated with the relative speed. This time is:\[ t = \frac{L}{v} = \frac{86.6 \text{ m}}{0.900c} = \frac{86.6}{0.9 \times 3 \times 10^8} \approx 3.21 \times 10^{-7} \text{ s} \].

Step 4: Determine If Paint Bomb Explodes (Train Frame)

Because the signal to deactivate the bomb is sent first (event RN occurs first), the bomb is deactivated before it can explode when the front of the train exits the tunnel. Hence, the paint bomb does not explode in the train's frame.

Step 5: Calculate Train Length in Tunnel Frame

From the tunnel’s perspective, the length of the moving train is contracted. The proper length of the train is given as 200 m. Calculating in the tunnel’s frame:\[ L' = L_0 \sqrt{1 - \frac{v^2}{c^2}} = 200 \times \sqrt{0.19} \approx 86.6 \text{ m} \].

Step 6: Determine Which Event Occurs First in Tunnel Frame

In the tunnel’s frame, the contracted train fits exactly within the 200 m long tunnel since both are perceived to be the same length. However, both events happen simultaneously in the tunnel’s frame; thus, both events FF and RN occur at the same time.

Step 7: Calculate Time Between Events in Tunnel Frame

In the tunnel’s frame, due to the simultaneity of both events occurring at once, the time duration between these two events is \( t = 0 \) seconds.

Step 8: Determine If Paint Bomb Explodes (Tunnel Frame)

Since both events happen simultaneously (the deactivation signal is received at the same time the front passes the end of the tunnel), the bomb does not explode as sufficient time isn't given for any sequential event.

Step 9: Resolve the Paradox

Both frames agree that the bomb does not explode. In the train frame, the deactivation signal is sent before the front exits the tunnel. In the tunnel frame, events RN and FF are simultaneous, thus extensive painting doesn't occur. Therefore, there's no paradox because all frames conclude that the paint bomb does not trigger.

Key Concepts

Length Contraction

Length contraction is a fundamental concept in the theory of relativity that occurs when objects are moving at speeds close to that of light. From the perspective of an observer in a different inertial frame, an object in motion appears shortened along the direction of motion. This is described by the formula:\[ L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \]Here, \( L_0 \) is the proper length, which is the length of the object in its rest frame (the frame in which it is not moving). Conversely, \( L \) is the contracted length as observed in a different inertial frame. In the example of the relativistic train, the proper length of both the train and tunnel is 200 meters. However, as the train moves through the tunnel at a velocity of 0.900c, the tunnel’s length as observed from the train's frame appears contracted to approximately 86.6 meters. Understanding length contraction helps in solving problems where high-speed motion affects the perceived dimensions of objects.

Simultaneity

Simultaneity in relativity refers to the concept that events perceived to occur at the same time in one frame of reference might not be simultaneous in another. This is significant because it opposes the Newtonian view of absolute time, suggesting that time is relative. In the problem involving the train and tunnel, we observe this principle when considering the sequence in which events occur. From the train's frame, event RN (rear of the train passing the near end of the tunnel) occurs before event FF (front of the train passing the far end of the tunnel). From the tunnel's frame, however, both events are perceived as happening simultaneously due to the train fitting within the contracted tunnel. Meaningful in resolving paradoxical scenarios, this concept underlies the conclusion that regardless of the perspective (train or tunnel frame), both agree on the outcome (the bomb not exploding), showing how simultaneity affects the interpretation of interactions and events.

Inertial Frames

An inertial frame is a frame of reference where objects obey Newton's first law of motion, remaining at rest or in uniform motion, unless acted upon by a force. In relativity, the concept of inertial frames is crucial for understanding how observations can differ based on perspective. In our example, two key inertial frames are considered: the train's frame and the tunnel's frame. In the train's frame, the train is at rest, and the tunnel moves at 0.900c. Conversely, in the tunnel's frame, the tunnel is at rest, and the train moves. As each frame offers its unique view on measurements such as time and length, understanding these frames helps in discussions about relativistic phenomena like length contraction and time dilation. Consistency in physical laws across different inertial frames makes outcomes (like the paint bomb not exploding) agree in all frames, harmonizing observations despite differing individual perspectives.

Time Dilation

Time dilation is an essential concept of relativity that illustrates how time passes differently for observers in different inertial frames, especially at high velocities. It occurs when a clock moving relative to an observer ticks slower compared to the observer's own clock. This is expressed by:\[ t = \frac{t_0}{\sqrt{1 - \frac{v^2}{c^2}}} \]Where \( t_0 \) is the proper time experienced in the rest frame of the clock, and \( t \) is the dilated time as observed from a different frame. In the train and tunnel scenario, time dilation affects how each frame perceives events and intervals between them. From the train's frame, the time interval between events RN and FF is stretched owing to the relatively higher speed, calculated as approximately 3.21 x 10^-7 seconds. However, in the tunnel's frame, the events occur with zero time between them, emphasizing the relativity of simultaneous events. Time dilation assures that whatever the frame, differences in time observations align with special relativity's predictions, ensuring coherent physical explanations across diverse frames.