Step 1: Conceptual Foundation:
This problem necessitates the computation of the de-Broglie wavelength for a macroscopic object. The de-Broglie hypothesis posits that all matter exhibits wave-like characteristics, with wavelength (\(\lambda\)) being inversely proportional to the object's momentum (\(p\)).
Step 2: Governing Equation:
The de-Broglie wavelength (\(\lambda\)) is determined by the following formula:
\[ \lambda = \frac{h}{p} = \frac{h}{mv} \]
where \(h\) represents Planck's constant (\(6.626 \times 10^{-34} \, \text{J}\cdot\text{s}\)), \(m\) denotes the mass of the object, and \(v\) signifies its velocity.
Step 3: Procedural Breakdown:
Provided Data:
Mass, \(m = 150 \, \text{g} = 0.150 \, \text{kg}\) (Conversion to SI units is mandatory).
Velocity, \(v = 30.0 \, \text{m/s}\).
Planck's constant, \(h = 6.626 \times 10^{-34} \, \text{J}\cdot\text{s}\).
Calculations:
Initially, calculate the momentum \(p\):
\[ p = mv = (0.150 \, \text{kg}) \times (30.0 \, \text{m/s}) = 4.5 \, \text{kg}\cdot\text{m/s} \]
Subsequently, compute the de-Broglie wavelength:
\[ \lambda = \frac{h}{p} = \frac{6.626 \times 10^{-34} \, \text{J}\cdot\text{s}}{4.5 \, \text{kg}\cdot\text{m/s}} \]
\[ \lambda \approx 1.4724 \times 10^{-34} \, \text{m} \]
Upon rounding, the result is \(1.47 \times 10^{-34} \, \text{m}\).
Step 4: Conclusive Result:
The de-Broglie wavelength of the ball is determined to be \(1.47 \times 10^{-34}\) m.