Problem 14 - University Physics with Modern Physics

Question

At rest, a rocket has an overall length of \(L .\) A garage at rest (built for the rocket by the lowest bidder) is only \(L / 2\) in length. Luckily, the garage has both a front door and a back door, so that when the rocket flies at a speed of \(v=0.866 c\), the rocket fits entirely into the garage. However, according to the rocket pilot, the rocket has length \(L\) and the garage has length \(L / 4\). How does the rocket pilot observe that the rocket does not fit into the garage?

Step-by-Step Solution

Answer

Explain your answer. Answer: No, the rocket does not fit into the garage from the rocket pilot's perspective. This is because, from the pilot's point of view, the garage is contracted to a length of L/4 due to length contraction in special relativity, while the rocket is still L in length. Since the contracted length of the garage is shorter than the length of the rocket, the rocket cannot fit into the garage from the pilot's perspective.

Step 1: Calculate the Lorentz factor

First, we need to find the Lorentz factor, which is given by the formula \[ \gamma = \dfrac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \] Plug in the given value of \(v = 0.866c\), we obtain, \[ \gamma = \dfrac{1}{\sqrt{1 - \frac{ (0.866c)^2}{c^2}}} = 2\]

Step 2: Calculate the contracted length of the rocket from the rocket pilot's perspective

According to the problem, the rocket pilot observes the rocket's length to be \(L\), which means there is no length contraction of the rocket from the pilot's perspective.

Step 3: Calculate the contracted length of the garage from the pilot's perspective

Length contraction formula is given by \[ L' = \dfrac{L}{\gamma} \] where \(L'\) is the contracted length, \(L\) is the original length, and \(\gamma\) is the Lorentz factor. As the garage has a length of \(L/2\) when it is at rest, the rocket pilot observes the garage to have a length of \[ L' = \dfrac{L/2}{\gamma} = \dfrac{L/2}{2} = \dfrac{L}{4}\]

Step 4: Determine whether the rocket fits into the garage

From the rocket pilot's perspective, the garage has a length of \(L/4\), while the rocket has a length of \(L\). Thus, the rocket does not fit into the garage from the pilot's point of view.

Step 5: Conclusion

In this exercise, we have used the concept of length contraction in special relativity to examine the situation where the rocket pilot perceives their rocket to not fit into the garage. By calculating the Lorentz factor and the contracted lengths of the rocket and garage, we've shown that from the rocket pilot's perspective, the rocket doesn't fit into the garage because the garage appears to be only \(L/4\) while the rocket is still \(L\).

Key Concepts

Lorentz factor

In the realm of special relativity, the Lorentz factor is crucial for understanding how time and space behave at high speeds. It's defined by the formula:

• \( \gamma = \dfrac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \)

This factor comes into play when objects travel at velocities close to the speed of light \((c)\), causing their time, length, and mass to transform relative to an observer at rest.
In our rocket problem, the speed \(v = 0.866c\) results in a Lorentz factor of 2. This means that from the perspective of an outside observer, time appears to slow down, and lengths contract in the direction of the motion. Understanding this factor helps us calculate how these quantities change, revealing the hidden dynamics of objects moving at relativistic speeds.

Length contraction

Length contraction is a fascinating consequence of special relativity. When an object moves at a significant fraction of the speed of light, its length in the direction of travel seems shorter to a stationary observer. This is given by the formula:

• \( L' = \dfrac{L}{\gamma} \)

Here, \(L'\) is the contracted length, \(L\) is the original length, and \(\gamma\) is the Lorentz factor.

In the exercise, the rocket's pilot sees no contraction of the rocket because they are moving with it, so its length remains \(L\). However, the garage appears shorter to the pilot due to its relative motion. Calculating with the Lorentz factor of 2, the garage's length contracts to \(L/4\) from the pilot's viewpoint.

This concept highlights how observers in different frames perceive different realities, each equally valid in their own context.

Rocket paradox

The rocket paradox elegantly demonstrates relativity's counterintuitive nature. From the stationary observer's frame, the rocket fits into the garage due to length contraction, giving the illusion of it being shorter.
However, from the rocket pilot's perspective, the garage is the object in motion, and thus it experiences length contraction, appearing only \(L/4\) long.

This paradox illustrates the core tenets of special relativity, where an observer in motion perceives the dimensions of stationary objects differently. It showcases how reality can differ substantively for observers in different frames, challenging our everyday intuitions about space and movement.

By exploring this paradox, we gain insight into the flexible and interconnected nature of time and space, where both perspectives coexist without contradiction in the relativistic world.