Questions: A 0.17 kg baseball thrown at 100 mph has a momentum of 7.7 kg · m/s. If the uncertainty in measuring the momentum is 1.0 × 10^-7 of the momentum, calculate the uncertainty in the baseball's position. Be sure your answer has the correct number of significant digits.

Solution Steps

We are given the momentum of a baseball and the uncertainty in measuring that momentum. We need to calculate the uncertainty in the baseball's position using the Heisenberg Uncertainty Principle.

The Heisenberg Uncertainty Principle is given by:

\[ \Delta x \cdot \Delta p \geq \frac{\hbar}{2} \]

where:

• \(\Delta x\) is the uncertainty in position,
• \(\Delta p\) is the uncertainty in momentum,
• \(\hbar\) is the reduced Planck's constant, \(\hbar = \frac{h}{2\pi} \approx 1.0546 \times 10^{-34} \, \text{J} \cdot \text{s}\).

The uncertainty in momentum \(\Delta p\) is given as \(1.0 \times 10^{-7}\) of the momentum. The momentum \(p\) is \(7.7 \, \text{kg} \cdot \text{m/s}\).

\[ \Delta p = 1.0 \times 10^{-7} \times 7.7 = 7.7 \times 10^{-7} \, \text{kg} \cdot \text{m/s} \]

Rearrange the Heisenberg Uncertainty Principle to solve for \(\Delta x\):

\[ \Delta x \geq \frac{\hbar}{2 \Delta p} \]

Substitute the known values:

\[ \Delta x \geq \frac{1.0546 \times 10^{-34}}{2 \times 7.7 \times 10^{-7}} \]

Calculate \(\Delta x\):

\[ \Delta x \geq \frac{1.0546 \times 10^{-34}}{1.54 \times 10^{-6}} \approx 6.8474 \times 10^{-29} \, \text{m} \]

Final Answer