Question

Imagine another universe in which the value of Planck's constant is 0.0663 J $\cdot$ s, but in which the physical laws and all other physical constants are the same as in our universe. In this universe, two physics students are playing catch. They are 12 m apart, and one throws a 0.25-kg ball directly toward the other with a speed of 6.0 m/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 125 cm${}^3$ at the time she throws it? (b) By what horizontal distance could the ball miss the second student?

Short answer

Momentum uncertainty is 0.1056 kg·m/s; possible miss distance is 0.8448 meters.

Step-by-step solution

Understanding the Problem and Given Values

We are working in a universe where the Planck's constant is given as 0.0663 J·s. We have a ball of mass 0.25 kg being thrown with a speed of 6.0 m/s, and it is known to be within a cube with a volume of 125 cm³. We need to find the uncertainty in momentum in a perpendicular direction and how this translates into a possible miss distance.

Convert Volume to Length

The volume of the cube is 125 cm³. We can find the length of one side of the cube by taking the cube root of the volume. Convert this side length from cm to meters (since other measurements are in meters):$$a=\sqrt[3]{125{\text{cm}}^3}=5\text{cm}=0.05\text{m}$$

Calculate Uncertainty in Position

Using one side of the cube as the uncertainty in position $\mathrm{\Delta }x$ (since it could be in any direction within the cube), we have:$$\mathrm{\Delta }x=0.05\text{m}$$

Apply Heisenberg's Uncertainty Principle

We apply the Heisenberg Uncertainty Principle to find the uncertainty in momentum $\mathrm{\Delta }p$ in the perpendicular direction:$$\mathrm{\Delta }x\mathrm{\Delta }p\ge \frac{h}{4\pi }$$Substitute $h=0.0663\text{J·s}$ and $\mathrm{\Delta }x=0.05\text{m}$:$$\mathrm{\Delta }p\ge \frac{0.0663}{4\pi \times 0.05}$$

Solve for Uncertainty in Momentum

Calculate the uncertainty in momentum:$$\mathrm{\Delta }p\ge \frac{0.0663}{0.628\text{m}}\approx 0.1056\text{kg·m/s}$$

Convert Uncertainty in Momentum to Miss Distance

Use the relationship between momentum uncertainty and velocity uncertainty to find the potential miss distance. The velocity uncertainty $\mathrm{\Delta }v=\mathrm{\Delta }p/m$:$$\mathrm{\Delta }v=\frac{0.1056\text{kg·m/s}}{0.25\text{kg}}=0.4224\text{m/s}$$Calculate the maximum horizontal miss distance over 12 m:$$\text{Miss distance}=\mathrm{\Delta }v\times t=\mathrm{\Delta }v\times \frac{12\text{m}}{6\text{m/s}}$$

Final Calculation for Miss Distance

Substitute $\mathrm{\Delta }v=0.4224\text{m/s}$ and calculate:$$\text{Miss distance}=0.4224\text{m/s}\times 2\text{s}=0.8448\text{m}$$

Conclusion

The uncertainty in the ball's horizontal momentum is approximately 0.1056 kg·m/s, and it could miss the second student by up to 0.8448 meters.

Key concepts

Planck's constant

In the world of quantum mechanics, Planck's constant, denoted as $h$, plays a pivotal role. It's a fundamental component in the Heisenberg Uncertainty Principle, representing the smallest quantum of action. In our universe, its value is approximately $6.626imes{10}^{-34}$ Joule-seconds. In the hypothetical universe of our problem, however, Planck's constant is a much larger $0.0663$ Joule-seconds. This larger value greatly affects the uncertainty in quantum measurements. The higher the Planck’s constant, the greater the inherent uncertainty in measurements of particles' positions and momenta. This means that quantum effects would be much more noticeable, even in larger objects like the ball from our problem. This constant essentially sets the scale at which quantum mechanical effects become significant. As such, increasing or decreasing its value alters the precision with which we can measure related properties like position and momentum.

momentum uncertainty

Momentum uncertainty is a core concept derived from the Heisenberg Uncertainty Principle, which can be expressed as $\mathrm{\Delta }x\mathrm{\Delta }p\ge \frac{h}{4\pi }$. Here, $\mathrm{\Delta }x$ is the uncertainty in position, and $\mathrm{\Delta }p$ is the uncertainty in momentum. In the given exercise, this principle came into play to determine how uncertain the momentum of a ball was when thrown between two students. By considering the ball's known position within a small cube, we applied this principle to find $\mathrm{\Delta }p$.

• $\mathrm{\Delta }p=\frac{h}{4\pi \mathrm{\Delta }x}$
• A larger uncertainty in position $\mathrm{\Delta }x$ would result in a smaller uncertainty in momentum $\mathrm{\Delta }p$, and vice versa.

In our exercise, using the larger theoretical value of Planck's constant, we find that the momentum uncertainty translated into a noticeable potential deviation in the ball's trajectory. This demonstrates how even a subtle change in Planck's constant can influence measurements and calculations in physics.

quantum mechanics

Quantum mechanics dives into the intriguing world of the very small, governing the behavior of particles at atomic and subatomic levels. It is a theory that challenges our classical intuition, stating that certain pairs of properties, such as position and momentum, cannot both be known to infinite precision.

• The Heisenberg Uncertainty Principle is a cornerstone of quantum mechanics.
• It highlights the probabilistic nature of quantum phenomena, suggesting that the act of measuring affects the system.
• This principle tells us that the more precisely a particle's position is known, the less precisely its momentum can be determined, and vice versa.

In our exercise, this principle was used to describe the situation of two students throwing a ball. Here, quantum mechanics teaches us about the limitations of measurements and how these become significant when dealing with very small scales, or when Planck's constant is larger, as in the provided scenario. Understanding these quantum effects helps us predict the behavior of systems on a microscopic scale, even affecting macroscopic events, such as a ball's possible trajectory deviation.