Problem 68 - College Physics

Question

\(\bullet\) In a parallel universe, the value of Planck's constant is 0.0663 \(\mathrm{J} \cdot\) 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 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

(a) Uncertainty in momentum: ~0.00529 kg·m/s. (b) Possible miss distance: ~0.529 m.

Step 1: Convert Volume Units

The volume of the cube where the ball is located is given as 1000 cm³. Convert this to cubic meters using the conversion factor (1 m = 100 cm):\[ 1000 \, \text{cm}^3 = 1000 \, \text{cm}^3 \times \left(\frac{1 \, m}{100 \, cm}\right)^3 = 1.0 \, \text{m}^3 \]

Step 2: Determine the Uncertainty in Position

Since the ball is located within a cube, each side of the cube is the cube root of the volume. The cube's side and hence the uncertainty in position, \( \Delta x \), is:\[ \Delta x = \sqrt[3]{1.0 \, \text{m}^3} = 1.0 \, \text{m} \]

Step 3: Use Heisenberg's Uncertainty Principle

Heisenberg's uncertainty principle states:\[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \]Where \( h \) is Planck's constant. In this universe, \( h = 0.0663 \, \text{J} \cdot \text{s} \). Substitute the values:\[ \Delta p \geq \frac{0.0663}{4\pi \cdot 1.0} \approx 0.00529 \, \text{kg} \cdot \text{m/s} \]

Step 4: Calculate the Uncertainty in Horizontal Distance Miss

The uncertainty in momentum \( \Delta p \) causes an uncertainty in the velocity. Since \( \Delta v = \frac{\Delta p}{m} \), where \( m = 0.10 \, \text{kg} \):\[ \Delta v = \frac{0.00529}{0.10} = 0.0529 \, \text{m/s} \]In the worst-case scenario, this velocity causes an additional horizontal distance miss over the 50 m throw:\[ \Delta x_{\text{miss}} = \Delta v \times t \]Since speed = distance/time, time \( t \) is \( \frac{50}{5} = 10 \) seconds:\[ \Delta x_{\text{miss}} = 0.0529 \, \text{m/s} \times 10 \, \text{s} = 0.529 \, \text{m} \]

Step 5: Conclusion

The uncertainty in the ball's horizontal momentum perpendicular to the direction of throw is approximately 0.00529 kg·m/s. The ball could miss the second student by roughly 0.529 meters due to this uncertainty.

Key Concepts

Planck's constant

Planck's constant is a crucial element in understanding quantum mechanics. It describes the size of the quantum effects in the quantum world. In the parallel universe described in the exercise, Planck's constant is given as 0.0663 J·s, which is different from the usual value in our universe, approximately 6.626 x 10⁻³⁴ J·s.

Planck's constant is pivotal in expressing the uncertainty in measurements, especially when dealing with very small particles. It establishes the limit to the precision with which we can simultaneously know quantities like position and momentum, leading us directly to Heisenberg’s Uncertainty Principle. This principle relies entirely on Planck's constant to set the bounds for our measurements.

Uncertainty in Position

In quantum physics, uncertainty in position (\(\Delta x\)) denotes how precisely we can pinpoint an object's location. In this exercise, the baseball is within a cube with a volume of 1000 cm³, which is converted into 1.0 m³ for easier calculations.

• To find the uncertainty in position, realize that the ball could be located anywhere along the cube's dimensions.
• If each dimension of the cube is equal, we calculate each side through the cube root of the volume.
  \[\Delta x = \sqrt[3]{1.0 \, \text{m}^3} = 1.0 \, \text{m}\]

This result shows considerable uncertainty in the ball's position given the size of the cube, reflecting a comprehensive impact on both motion and subsequent calculations.

Momentum

Momentum is the product of an object's mass and velocity, given as \( p = mv \). In context, we're concerned with the uncertainty of momentum (\( \Delta p \)).

According to Heisenberg's Uncertainty Principle:

• \[\Delta x \cdot \Delta p \geq \frac{h}{4\pi}\]
• Here, \( \Delta p \) represents the smallest uncertainty in momentum we can measure based on position uncertainty \( \Delta x \).
• For the exercise's alternate universe, determining \( \Delta p = 0.00529 \, \text{kg} \cdot \text{m/s} \) implies that the student's ability to predict the ball's movement trajectory is constrained by unavoidably imprecise momentum measurements.

This concept captures the core of quantum mechanics by limiting our ability to know both position and momentum precisely.

Converting Units

Converting units is fundamental in science, ensuring measurements are easy to apply. Proper conversions allow seamless integration of different unit systems and accurate interpretations of physical laws.

Here's how converting units played a crucial role in the given problem:

• The original volume was in 1000 cm³, which needed conversion to cubic meters:
  • \[1000 \, \text{cm}^3 = 1000 \, \text{cm}^3 \times \left(\frac{1 \, \text{m}}{100 \, \text{cm}}\right)^3 = 1.0 \, \text{m}^3\]
• This conversion impacts calculations of uncertainty in position \( \Delta x \) by standardizing the unit system for the volume.
• Missteps in unit conversions could lead to significant errors, influencing outcomes such as trajectory estimations or force measurements.

Effective unit conversions assure accuracy and avoid confusion, critical in complex scientific computations like these.