The two paricles A and B have de Broglie wavelengths 1 nm and 5 nm respectively if mass of A is four times the mass of B, the ratio of kinetic energies of A and B would be :-

Options

A. `(a)5 : 1`
B. (b)`25 : 4`
C. (c)`20 : 1`
D. (d)`5 : 4`

Answer

To solve the problem, we need to find the ratio of the kinetic energies of particles A and B given their de Broglie wavelengths and the relationship between their masses.

### Step-by-Step Solution:

1. **Identify Given Data:**
- De Broglie wavelength of particle A, \( \lambda_A = 1 \, \text{nm} \)
- De Broglie wavelength of particle B, \( \lambda_B = 5 \, \text{nm} \)
- Mass of particle A, \( m_A = 4m_B \) (where \( m_B \) is the mass of particle B)

2. **Use the de Broglie Wavelength Formula:**
The de Broglie wavelength is given by the formula:
\[ \lambda = \frac{h}{\sqrt{2mK}} \]
where \( h \) is Planck's constant, \( m \) is the mass of the particle, and \( K \) is the kinetic energy.

3. **Set Up Equations for Both Particles:**
For particle A:
\[ \lambda_A = \frac{h}{\sqrt{2m_A K_A}} \implies 1 = \frac{h}{\sqrt{2(4m_B)K_A}} \quad \text{(1)} \]
For particle B:
\[ \lambda_B = \frac{h}{\sqrt{2m_B K_B}} \implies 5 = \frac{h}{\sqrt{2m_B K_B}} \quad \text{(2)} \]

4. **Cross-Multiply and Rearrange:**
From equation (1):
\[ \sqrt{2(4m_B)K_A} = h \implies 2(4m_B)K_A = h^2 \implies 8m_B K_A = h^2 \implies K_A = \frac{h^2}{8m_B} \]
From equation (2):
\[ \sqrt{2m_B K_B} = \frac{h}{5} \implies 2m_B K_B = \left(\frac{h}{5}\right)^2 \implies 2m_B K_B = \frac{h^2}{25} \implies K_B = \frac{h^2}{50m_B} \]

5. **Calculate the Ratio of Kinetic Energies:**
Now, we can find the ratio \( \frac{K_A}{K_B} \):
\[ \frac{K_A}{K_B} = \frac{\frac{h^2}{8m_B}}{\frac{h^2}{50m_B}} = \frac{50}{8} = \frac{25}{4} \]

6. **Final Answer:**
The ratio of the kinetic energies of particles A and B is:
\[ \frac{K_A}{K_B} = \frac{25}{4} \]