Problem 69 - Fundamentals of Physics

Question

The car-in-the-garage problem. Carman has just purchased the world's longest stretch limo, which has a proper length of \(L_{c}=30.5 \mathrm{~m} .\) In Fig. \(37-32 a,\) it is shown parked in front of a garage with a proper length of \(L_{g}=6.00 \mathrm{~m}\). The garage has a front door (shown open) and a back door (shown closed). The limo is obviously longer than the garage. Still, Garageman, who owns the garage and knows something about relativistic length contraction, makes a bet with Carman that the limo can fit in the garage with both doors closed. Carman, who dropped his physics course before reaching special relativity, says such a thing, even in principle, is impossible. To analyze Garageman's scheme, an \(x_{c}\) axis is attached to the limo, with \(x_{c}=0\) at the rear bumper, and an \(x_{k}\) axis is attached to the garage, with \(x_{g}=0\) at the (now open) front door. Then Carman is to drive the limo directly toward the front door at a velocity of \(0.9980 c\) (which is, of course, both technically and financially impossible). Carman is stationary in the \(x_{c}\) reference frame; Garageman is stationary in the \(x_{z}\) reference frame. There are two events to consider. Event 1 : When the rear bumper clears the front door, the front door is closed. Let the time of this event be zero to both Carman and Garageman: \(t_{\mathrm{s} 1}=t_{c 1}=0\). The event occurs at \(x_{c}=x_{g}=0 .\) Figure \(37-32 b\) shows event 1 according to the \(x_{g}\) reference frame. Event 2 : When the front bumper reaches the back door, that door opens. Figure \(37-32 c\) shows event 2 according to the \(x_{g}\) reference frame. According to Garageman, (a) what is the length of the limo, and what are the spacetime coordinates (b) \(x_{R 2}\) and (c) \(t_{g 2}\) of event \(2 ?\) (d) For how long is the limo temporarily "trapped" inside the garage with both doors shut? Now consider the situation from the \(x_{c}\) reference frame, in which the garage comes racing past the limo at a velocity of \(-0.9980 \mathrm{c}\). According to Carman, (e) what is the length of the passing garage, what are the spacetime coordinates (f) \(x_{c 2}\) and \((g) t_{c 2}\) of event \(2,(h)\) is the limo ever in the garage with both doors shut, and (i) which event occurs first? (j) Sketch events 1 and 2 as seen by Carman. (k) Are the events causally related; that is, does one of them cause the other? (I) Finally, who wins the bet?

Step-by-Step Solution

Answer

The limo is temporarily trapped from the garage's frame, but never from the limo's frame, so technically Carman wins the bet.

Step 1: Understand the Lorentz Transformation

In this problem, we are working in two different reference frames: one attached to the garage and one attached to the limo. Due to the high velocity involved, we will need to apply the Lorentz transformation to understand length contraction and time dilation effects. The basic form of the Lorentz transformation for length contraction is given by: \[ L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \]where \( L_0 \) is the proper length (in the rest frame of the object), \( v \) is the velocity of the object relative to the observer, and \( c \) is the speed of light.

Step 2: Calculate the Length of the Limo in Garage's Frame

According to Garageman's frame, we need to calculate the relativistically contracted length of the limo. The limo's proper length is \( L_c = 30.5 \ m \), and it travels at \( 0.9980c \). Substitute these values into the length contraction formula:\[ L = L_0 \sqrt{1 - \frac{v^2}{c^2}} = 30.5 \cdot \sqrt{1 - (0.9980)^2} \]Calculate the contraction:\[ L = 30.5 \cdot \sqrt{1 - 0.996004} \approx 30.5 \cdot 0.0632 \approx 1.93 \ m \]So, according to Garageman, the length of the limo is \( 1.93 \ m \).

Step 3: Determine Spacetime Coordinates in Garage's Frame

To find the spacetime coordinates for event 2 in the garage's frame (when the front bumper reaches the back door), note this is equivalent to the front of the limo reaching the back of the garage. Event 1 is set at \( t_{g1} = 0 \) and \( x_{g1} = 0 \). The limo's front reaches the back door of the garage at \( x_{g2} = L_g = 6 \ m \) since the length of the limo in the garage frame is shorter than the garage.The time coordinate can be found by considering that the limo travels this contracted length \( L \,\text{(1.93 m)} \) at \( 0.998c \):\[ t_{g2} = \frac{x_{g2} - x_{g1}}{v} = \frac{6 \ m}{0.998c} \approx 2.007 \times 10^{-8} \ s \]

Step 4: Calculate the "Trapped" Duration in Garage's Frame

The limo is 'trapped' in the sense that both doors of the garage are closed. This happens between event 1 and when the front bumper exits the back of the garage, which is event 2. We previously calculated \( t_{g2} \) as the time taken for the entire limo to be inside the garage.Since the rear bumper is outside at \( t_{g1} = 0 \), and the front bumper leaves at \( t_{g2} \), the limo is temporarily trapped for:\[ \Delta t_{trapped} = t_{g2} - t_{g1} = 2.007 \times 10^{-8} \, s \]

Step 5: Consider the Perception from the Limo's Frame

Switching to Carman's frame, the garage moves towards the limo at \(-0.998c\). In this frame, calculate the length contraction of the garage with its proper length \(L_g = 6 \, m\):\[ L' = L_g \sqrt{1 - \frac{v^2}{c^2}} = 6 \cdot \sqrt{1 - 0.996004} \approx 6 \cdot 0.0632 \approx 0.38 \, m \]According to Carman, the length of the passing garage is \( 0.38 \, m \).

Step 6: Find Spacetime Coordinates in Limo's Frame

In Carman's frame, we now find spacetime coordinates for when the garage's front hits the limo's back at \( x_{c2} \). Event 1 occurs when the garage's back door reaches the limo's front in her frame.Given the garage's contracted length equals \( x_{c2} = x_c2 = L' = 0.38 \ m \).Estimate time using \( t_{c2} \), considering the garage length \( L' \) and velocity:\[ t_{c2} = \frac{L'}{v} = \frac{0.38 \, m}{0.998c} \approx \approx 1.27 \times 10^{-9} \, s \]

Step 7: Analyze Events and Determine Bet Winner

In Carman's frame, due to length contraction, the entire garage never encloses the limo. Thus, from Carman's perspective, the limo is never trapped. Therefore, Garageman's claim of temporally trapping the limo in the garage does not hold in the limo’s frame. Conclusively, since space-time is relative, neither frame causes the other, and Garageman doesn't win conclusively from all perspectives.

Key Concepts

Length Contraction

Length contraction is a key concept in relativity that emerges from Einstein's special theory of relativity. It refers to the phenomenon where an object appears shorter along the direction of motion when it is moving at a significant fraction of the speed of light, relative to an observer. This contraction only occurs in the direction of travel. The equation used to determine the contracted length is:\[ L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \]Here,

• \( L \) is the contracted length as observed in the moving reference frame,
• \( L_0 \) is the proper length or rest length of the object,
• \( v \) is the velocity of the object relative to the observer, and
• \( c \) is the speed of light.

Applying this to the car-in-the-garage problem, Garageman, viewing the limo moving past his garage at 0.9980c, observes the limo as being shorter than its proper length (30.5 m). By substituting the values into the formula, the limo is contracted to approximately 1.93 meters in his reference frame. This surprising effect makes it seem possible for a longer object to fit into a shorter space when viewed at high speeds.

Lorentz Transformation

The Lorentz transformation is an essential tool in relativistic physics, allowing us to connect different observations from different reference frames. It describes how the measurements of time and space are related for observers in different frames in relative motion, particularly at velocities close to the speed of light. These transformations account for the observed phenomena of length contraction and time dilation. The transformations can be described by the following equations for space (\(x\)) and time (\(t\)):

• For space: \[ x' = \gamma (x - vt) \]
• For time: \[ t' = \gamma \left(t - \frac{vx}{c^2}\right) \]

Where,

• \( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \) is the Lorentz factor, which increases with velocity,
• \( x \) and \( t \) are the coordinates in one frame (e.g., stationary reference), and
• \( x' \) and \( t' \) are the coordinates in the moving reference frame.

Using these equations, Garageman and Carman can calculate how the events and measurements differ in their respective frames. Even though the events will be observed differently, the Lorentz transformation ensures the physical laws stay consistent in all inertial frames.

Reference Frames

Reference frames are essential to understanding relativity as they provide context and perspective for observers to measure events. A reference frame is essentially a viewpoint with a coordinate system to specify where and when events occur. In the relative scenario of the car-in-the-garage problem, two reference frames are considered:

• The garage's reference frame, where the garage appears stationary and the limo approaches at 0.9980c. In this frame, length contraction makes the limo appear shorter so it can fit in the garage.
• The limo's reference frame, where the limo is at rest and the garage moves towards it. In this frame, the garage experiences length contraction, making it seem much shorter than the limo.

Different frames can lead to different measurements and interpretations of the same event, highlighting that observations depend significantly on the observer's state of motion. By analyzing both frames, one can better understand how each observer sees the sequence of events.

Time Dilation

Time dilation is another fascinating result of the special theory of relativity, which suggests that time can pass at different rates for observers in different frames of motion. It indicates that a clock in a moving system ticks slower compared to a clock in a stationary frame. The formula for time dilation is:\[ \Delta t = \frac{\Delta t_0}{\sqrt{1 - \frac{v^2}{c^2}}} \]Here,

• \( \Delta t \) is the dilated time interval observed in the stationary frame,
• \( \Delta t_0 \) is the proper time as measured in the moving frame, and
• \( v \) and \( c \) denote the velocity and speed of light, respectively.

For Garageman and Carman, each sees time passing differently due to their relative motion. In the garage's frame, it appears as if the limo's clock runs slower, leading to a perception of time dilation. These differences in time perception help explain why events seem to unfold differently depending on the reference frame, such as how long the limo is trapped inside the garage. Understanding time dilation is crucial to realizing why the same event can appear so distinct in separate frames.