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 \(x_{g}\) 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 \(0.9980 \mathrm{c}\) (which is, of course, both technically and financially impossible). Carman is stationary in the \(x_{c}\) reference frame; Garageman is stationary in the \(x_{g}\) 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: \(t_{g 1}=t_{c 1}=0\). 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 \(\bar{x}_{8}\) reference frame. According to Garageman, (a) what is the length of the limo, and what are the spacetime coordinates (b) \(x_{g^{2}}\) 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 \(-0.9980 c\). 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 \(2,(\mathrm{~h})\) 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 fits inside due to length contraction in Garageman's frame, and Garageman wins the bet. Events are not causally linked.

Step 1: Calculate Relativistic Length Contraction

The first step is to find the contracted length of the limo from Garageman's perspective. The formula for relativistic length contraction is given by \( L' = L \sqrt{1 - \frac{v^2}{c^2}} \). Here, the proper length \( L = 30.5 \text{ m} \), and the speed \( v = 0.9980c \). Substituting these values, we get \( L' = 30.5 \times \sqrt{1 - (0.9980)^2} = 1.52 \text{ m} \). Therefore, the relativistically contracted length of the limo is 1.52 m as observed by Garageman.

Step 2: Calculate Spacetime Coordinate \(x_{g2}\)

Since event 2 occurs when the front bumper reaches the back door, it happens at a distance equal to the proper length of the garage from the front door of the garage. Thus, \(x_{g2} = L_g = 6.00 \text{ m}\).

Step 3: Calculate Time \(t_{g2}\) of Event 2

To find the time \(t_{g2}\) at which event 2 occurs, we can use the relation \(v = \frac{x_{g2}}{t_{g2}}\). Substituting for \(x_{g2}\) and \(v\), we get \(t_{g2} = \frac{6.00}{0.9980c} \approx 2.00 \times 10^{-8} \text{ s} \) where \( c \) is the speed of light.

Step 4: Determine Time Duration Limo is in Garage

The limo is trapped inside the garage from the time the rear enters until the front exits. Since \(L' = 1.52 \, \text{m}\), both doors are closed for a time given by \( \Delta t = \frac{L' - L_g}{v} \), which evaluates to a very small time, essentially 0 since \(L'\) is smaller than \(L_g\).

Step 5: Calculate Length of Passing Garage (Carman's Frame)

In Carman's reference frame, the garage experiences relativistic length contraction. The contracted length \(L'_g = L_g \sqrt{1 - \frac{v^2}{c^2}} = 6.00 \times \sqrt{1 - (0.9980)^2} = 0.299 \text{ m}\).

Step 6: Calculate Spacetime Coordinate \(x_{c2}\)

From Carman's perspective, event 2 coordinates \(x_{c2}\) can be determined through the length of the limo. Since Carman sees the garage moving, \(x_{c2} = - (L_{c} - x_{c1}) \), approximately \(-24.5 \text{ m}\).

Step 7: Determine Time \(t_{c2}\) of Event 2

Using the relativistic speed \(v\), find time \(t_{c2} \) in Carman's frame, equivalent to the contracted garage length passing time. Utilizing Lorentz transformation,\( t_{c2} = \frac{x_{c2}}{-v} \). The result matches the earlier calculation converted for Carman's perception.

Step 8: Evaluate Sequence and Causal Relation of Events

In Carman's frame, event 1 precedes due to relativistic effects, meaning both physical interpretations are true in respective inertial frames. Events 1 and 2 do not share a causal relation strictly due to relativistic independence.

Step 9: Determine Bet Outcome

Garageman wins due to relativistic length contraction from his perspective, temporarily trapping the limo inside the garage.

Step 10: Finalize Sketch Description

Sketch order showing event 1 arrival and then event 2 departure. Highlight sequence as interpreted differently in each frame, showcasing relativity impact.

Key Concepts

Special Relativity

Special relativity is a theory formulated by Albert Einstein that revolutionized our understanding of physics, particularly in terms of motion and measurement. It primarily deals with objects moving at speeds close to the speed of light. One of its key postulates is that the laws of physics are identical in all inertial reference frames, which means that whether you're standing still or moving at a constant velocity, the physics do not change.

Another crucial aspect of special relativity is that the speed of light remains constant, irrespective of the observer's motion. This forms the basis for many relativistic phenomena. For instance, time dilation and length contraction are direct results. Length contraction, which is central to our car-in-the-garage problem, posits that objects will appear shorter in the direction of motion when observed from a different inertial frame moving relative to the object.

In simpler terms, as the limo accelerates to nearly the speed of light, its length appears contracted to the garageman due to the high velocity (\(v = 0.9980c\)), resulting in an apparent length of just 1.52 m, allowing it to fit inside the shorter garage. This concept highlights the counterintuitive nature of special relativity by challenging our everyday perceptions of space and time.

Spacetime Coordinates

Spacetime coordinates are a fundamental concept within relativity, combining the three dimensions of space with the single dimension of time into a four-dimensional spacetime continuum. In this framework, any event that occurs is pin-pointed by a unique set of spacetime coordinates, consisting of spatial location coordinates (\(x, y, z\)) and a time coordinate (\(t\)).

In the car-in-the-garage problem, spacetime coordinates help track the occurrence of key events, like the opening and closing of the garage doors. From each observer’s perspective, events are measured differently not only spatially but temporally too. For Garageman, when the rear bumper enters the garage, we assign it coordinates (\(t_{g1} = 0\), \(x_{g} = 0\)), and when the front bumper reaches the back, we have (\(x_{g2} = 6.00 \, \text{m}\), \(t_{g2} = 2.00 \times 10^{-8} \, \text{s}\)).

This illustrates how spacetime coordinates allow for a comprehensive understanding of events across different frames of reference, accommodating for the relative perception of time and space stretches and contractions.

• Space coordinates: define locations in space.
• Time coordinate: marks the instant an event occurs.
• Spacetime coordinates: unify space and time for event tracking.

Inertial Reference Frames

Inertial reference frames are systems where objects move at constant velocity when no external forces are acting on them. Within these frames, Newton's laws of motion hold true, and all spatial directions are equivalent. In special relativity, each observer is considered to be in their own inertial frame if they are not accelerating.

In our scenario with Carman and Garageman, each has their own inertial reference frame. Carman, driving the limo, considers himself stationary, observing the garage as moving towards him. Conversely, Garageman observes the limo as moving towards him. This difference highlights the relativity of motion: what appears fixed from one's point of view may be moving from another's.

Understanding these reference frames is vital for grasping relativistic effects like length contraction. Each observer’s measurements of space and time are correct within their own frame but will differ from the measurements made in another frame. This relativity of simultaneity underlines why both Carman and Garageman perceive the events of the limo’s entrance and exit from the garage differently.

• Stationary perspective: each observer feels as though they are not moving.
• Relative motion: both frames perceive the movement of others.
• Relativity principle: physics laws hold the same in all inertial frames.