Problem 63 - College Physics

Question

In a parallel universe, the value of Planck's constant is \(0.0663 \mathrm{~J} \cdot \mathrm{s}\). Assume that the physical laws and all other physical constants are the same as in our universe. In this other universe, two physics students are playing catch with a baseball. They are \(50 \mathrm{~m}\) apart, and one throws a \(0.10 \mathrm{~kg}\) ball with a speed of \(5.0 \mathrm{~m} / \mathrm{s}\). (a) What is the uncertainty in the ball's horizontal momentum in a direction perpendicular to that in which it is being thrown if the student throwing the ball knows that it is located within a cube with volume \(1000 \mathrm{~cm}^{3}\) at the time she throws it? (b) By what horizontal distance could the ball miss the second student?

Step-by-Step Solution

Answer

Momentum uncertainty: \(0.0528 \text{ kg} \cdot \text{m/s}\). Max deviation: \(5.28 \text{ m}\).

Step 1: Understanding the Problem

We are given a parallel universe where Planck's constant is different, and we need to compute the uncertainty in the ball’s horizontal momentum given certain conditions. Let's use Heisenberg's uncertainty principle for our calculation.

Step 2: Identify Known Variables

Planck's constant in this universe is \(h = 0.0663 \text{ J} \cdot \text{s}\). The volume of the cube is \(1000 \text{ cm}^3\), which means the length of each side is \(10 \text{ cm}\) or \(0.1 \text{ m}\). The mass of the ball is \(0.10 \text{ kg}\), and it is thrown at \(5.0 \text{ m/s}\).

Step 3: Calculate Uncertainty in Position

Since the ball is located within a cube, the uncertainty in position, \(\Delta x\), can be assumed as the edge length of the cube: \(\Delta x = 0.1 \text{ m}\).

Step 4: Applying Heisenberg's Uncertainty Principle

Heisenberg's uncertainty principle states that \(\Delta x \cdot \Delta p \geq \frac{h}{4\pi}\). Plug in the values: \(0.1 \cdot \Delta p \geq \frac{0.0663}{4\pi}\) to find \(\Delta p\).

Step 5: Solve for Momentum Uncertainty

Calculate \(\Delta p\):\[\Delta p \geq \frac{0.0663}{4\pi \cdot 0.1} \approx 0.0528 \text{ kg} \cdot \text{m/s}\]

Step 6: Calculate Maximum Deviation

Use the formula \( \Delta x_{final} = \frac{\Delta p \cdot L}{m \cdot v} \). Here, \(L = 50\text{ m}\), \(m = 0.1\text{ kg}\), and \(v = 5.0\text{ m/s}\).Substitute the known values:\[\Delta x_{final} = \frac{0.0528 \cdot 50}{0.10 \cdot 5.0} = 5.28 \text{ m}\]

Step 7: Conclusion

The uncertainty in the ball's horizontal momentum perpendicular to its direction is approximately \(0.0528 \text{ kg} \cdot \text{m/s}\). The ball could miss the student by as much as \(5.28 \text{ m}\) horizontally.

Key Concepts

Planck's Constant

Planck's constant is a fundamental value in physics, symbolized as \( h \), that helps in understanding the quantum mechanics realm. It is usually approximately \( 6.626 imes 10^{-34} \text{ J} \cdot \text{s} \) in our universe. However, in this parallel universe exercise, it is set to a much larger value of \( 0.0663 \text{ J} \cdot \text{s} \). This change significantly affects calculations related to quantum mechanics.

Planck's constant governs the scale at which quantum mechanical effects appear. The larger the constant, the more pronounced such effects are in everyday macroscopic objects. This means that phenomena like the uncertainty principle become more observable even in larger objects, like a baseball. This adjustment sets the scene for understanding how the fundamental laws of physics might function differently in a universe with an altered Planck's constant. Therefore, tasks that seem routine in our world, such as playing catch, can become interesting exercises in quantum uncertainty.

Momentum Uncertainty

Momentum uncertainty refers to the imprecision or lack of perfect predictability about a particle's momentum, a core aspect of Heisenberg’s Uncertainty Principle. Here, the principle relates the uncertainties of position (\( \Delta x \)) and momentum (\( \Delta p \)).

Given by the formula \( \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \), this principle shows that increasing precision in measuring a particle's position results in increased uncertainty in measuring its momentum. In our exercise, the baseball's position uncertainty is set by the cube it is confined within, \( 0.1 \text{ m} \), leading to the calculation of momentum uncertainty. This results in a \( \Delta p \approx 0.0528 \text{ kg} \cdot \text{m/s} \).

Understanding this is crucial to predicting how various factors influence quantum behavior in macroscopic objects. The larger value of Planck's constant in this universe amplifies this uncertainty, showcasing how intertwined measurements are in quantum mechanics.

Physics in Parallel Universes

The concept of parallel universes introduces an exercise not just in imagination but also in understanding physics under altered conditions. In this scenario, we explore the universe where Planck’s constant differs from our own. By seeing how changes in fundamental constants can impact the laws of physics, we gain insight into consistency and variability across potential realities.

Understanding these differences can challenge assumptions about how natural laws might universally apply. It poses interesting questions regarding what realities, similar or vastly different, might exist with variations in constants like Planck’s constant. This study reveals how principles we take for granted could shift, shedding light on the complexity and flexibility of the quantum mechanical framework we exist in.

Quantum Mechanics in Macroscopic Objects

Quantum mechanics is often associated with small-scale phenomena at the atomic and subatomic levels. However, in a universe with an altered Planck's constant, quantum properties extend their influence to larger, macroscopic objects like baseballs. This highlights the incredible and often surprising ways in which quantum mechanics can impact what we perceive as the classical world.

When Planck’s constant is large, objects normally considered too big to exhibit quantum mechanical behavior start demonstrating these effects, which in our universe are typically too minor to detect. For instance, a baseball might not land where expected due to significant uncertainty. Such exercises expand our understanding of quantum mechanics beyond its usual domain and prompt us to reconsider how confidentiality and randomness might integrate into ordinary experiences across different universes.