Find the de-Broglie wavelengths of (a) a 46 g golf ball with a velocity of 30m/s (b) an electron with a velocity of `10^7 m//s.`

Explanation

To find the de-Broglie wavelengths for the given objects, we will use the de-Broglie wavelength formula:

\[ \lambda = \frac{h}{mv} \]

where:
- \(\lambda\) is the de-Broglie wavelength,
- \(h\) is the Planck's constant (\(6.63 \times 10^{-34} \, \text{J s}\)),
- \(m\) is the mass of the object in kilograms,
- \(v\) is the velocity of the object in meters per second.

### Part (a): Golf Ball

1. **Convert mass of the golf ball to kilograms:**
\[ m = 46 \, \text{g} = 0.046 \, \text{kg} \]

2. **Use the given velocity:**
\[ v = 30 \, \text{m/s} \]

3. **Substitute the values into the de-Broglie wavelength formula:**
\[ \lambda = \frac{6.63 \times 10^{-34} \, \text{J s}}{0.046 \, \text{kg} \times 30 \, \text{m/s}} \]

4. **Calculate the denominator:**
\[ mv = 0.046 \, \text{kg} \times 30 \, \text{m/s} = 1.38 \, \text{kg m/s} \]

5. **Calculate the wavelength:**
\[ \lambda = \frac{6.63 \times 10^{-34}}{1.38} \approx 4.8 \times 10^{-34} \, \text{m} \]

### Part (b): Electron

1. **Use the mass of the electron:**
\[ m = 9.1 \times 10^{-31} \, \text{kg} \]

2. **Use the given velocity:**
\[ v = 10^7 \, \text{m/s} \]

3. **Substitute the values into the de-Broglie wavelength formula:**
\[ \lambda = \frac{6.63 \times 10^{-34} \, \text{J s}}{9.1 \times 10^{-31} \, \text{kg} \times 10^7 \, \text{m/s}} \]

4. **Calculate the denominator:**
\[ mv = 9.1 \times 10^{-31} \, \text{kg} \times 10^7 \, \text{m/s} = 9.1 \times 10^{-24} \, \text{kg m/s} \]

5. **Calculate the wavelength:**
\[ \lambda = \frac{6.63 \times 10^{-34}}{9.1 \times 10^{-24}} \approx 7.3 \times 10^{-11} \, \text{m} \]

### Final Answers:
- (a) The de-Broglie wavelength of the golf ball is approximately \(4.8 \times 10^{-34} \, \text{m}\).
- (b) The de-Broglie wavelength of the electron is approximately \(7.3 \times 10^{-11} \, \text{m}\).