What is the de Broglie wavelength corresponding to a ball of mass 100 g moving with a speed of 33 m/s? (Plank's constant-6.6 x10-34 J/s)
What is the de Broglie wavelength corresponding to a ball of mass 100 g moving with a speed of 33 m/s? (Plank's constant-6.6 x10-34 J/s)
1x10-34 m
2x10-34 m
3x10-34 m
1x1034 m
2x1034 m
The Correct Option is B
Approach Solution - 1
Given:
- Mass of the ball, \( m = 100 \, \text{g} = 0.1 \, \text{kg} \)
- Speed of the ball, \( v = 33 \, \text{m/s} \)
- Planck's constant, \( h = 6.6 \times 10^{-34} \, \text{J} \cdot \text{s} \)
Step 1: De Broglie Wavelength Formula
The de Broglie wavelength (\( \lambda \)) is given by:
\[ \lambda = \frac{h}{p} = \frac{h}{m v} \]
where \( p \) is the momentum of the object.
Step 2: Substitute the Given Values
\[ \lambda = \frac{6.6 \times 10^{-34}}{0.1 \times 33} \]
\[ \lambda = \frac{6.6 \times 10^{-34}}{3.3} \]
\[ \lambda = 2 \times 10^{-34} \, \text{m} \]
Conclusion:
The de Broglie wavelength corresponding to the ball is \( 2 \times 10^{-34} \, \text{m} \).
Answer: \(\boxed{B}\)
Approach Solution -2
Step 1: Recall the formula for de Broglie wavelength.
The de Broglie wavelength (\( \lambda \)) of a particle is given by:
\[ \lambda = \frac{h}{p}, \]
where:
- \( h \) is Planck's constant ( \(h = 6.6 \times 10^{-34} \, \text{J.s}\)),
- \( p \) is the momentum of the particle, given by \( p = mv \),
- \( m \) is the mass of the particle, and
- \( v \) is the velocity of the particle.
We are given:
- \( m = 100 \, \text{g} = 0.1 \, \text{kg}, \)
- \( v = 33 \, \text{m/s}, \)
- \( h = 6.6 \times 10^{-34} \, \text{Js}. \)
---
Step 2: Calculate the momentum (\( p \)).
The momentum of the ball is:
\[ p = mv = 0.1 \times 33 = 3.3 \, \text{kgm/s}. \]
---
Step 3: Substitute into the de Broglie wavelength formula.
Substitute \( h = 6.6 \times 10^{-34} \, \text{J·s} \) and \( p = 3.3 \, \text{kgm/s} \):
\[ \lambda = \frac{h}{p} = \frac{6.6 \times 10^{-34}}{3.3}. \]
Simplify:
\[ \lambda = 2 \times 10^{-34} \, \text{m}. \] ---
Final Answer: The de Broglie wavelength is \( \mathbf{2 \times 10^{-34} \, \text{m}} \), which corresponds to option \( \mathbf{(B)} \).
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