Question

Two base balls(masses; m₁=100g, and m₂=50g) are thrown. Both move with uniform velocity, but the velocity of m₂ is 1.5 times that of m₁.The ration of de Broglie wavelengths λ(m₁) : λ(m₂) is given by

• A. 4:3
• B. 3:4
• C. 2:1
• D. 1:2

Answer 1

The Correct Option is B

 de-Broglie Wavelength Ratio 

Given:
$m_1 = 100 \text{ g}, \quad V_1 = V_1 \text{ m/s}$
$m_2 = 50 \text{ g}, \quad V_2 = 1.5V_1 \text{ m/s}$

To find:
$\dfrac{\lambda(m_1)}{\lambda(m_2)} = ?$

Using de-Broglie’s wavelength formula:
$\lambda = \dfrac{h}{mv}$

So, the ratio is: \[ \dfrac{\lambda_1}{\lambda_2} = \dfrac{m_2 V_2}{m_1 V_1} = \dfrac{50 \times 1.5V_1}{100V_1} = \dfrac{3}{4} \]

Final Answer: The ratio is 3 : 4.

Answer 2

The de Broglie wavelength ($\lambda$) of a particle is given by the formula:

$\lambda = \frac{h}{p} = \frac{h}{mv}$

where: 

• $h$ is Planck's constant
• $m$ is the mass of the particle
• $v$ is the velocity of the particle

We are given two baseballs with masses $m_1 = 100 \text{ g}$ and $m_2 = 50 \text{ g}$. The velocity of $m_2$ is 1.5 times that of $m_1$, so $v_2 = 1.5v_1$.

We want to find the ratio of their de Broglie wavelengths: $\lambda(m_1) : \lambda(m_2)$

The de Broglie wavelength for mass $m_1$ is:

$\lambda_1 = \frac{h}{m_1v_1}$

The de Broglie wavelength for mass $m_2$ is:

$\lambda_2 = \frac{h}{m_2v_2}$

Now, let's find the ratio $\frac{\lambda_1}{\lambda_2}$:

$\frac{\lambda_1}{\lambda_2} = \frac{\frac{h}{m_1v_1}}{\frac{h}{m_2v_2}} = \frac{m_2v_2}{m_1v_1}$

Substitute the given values: $m_1 = 100$, $m_2 = 50$, and $v_2 = 1.5v_1$:

$\frac{\lambda_1}{\lambda_2} = \frac{50 \cdot 1.5v_1}{100 \cdot v_1} = \frac{75}{100} = \frac{3}{4}$

Therefore, the ratio of the de Broglie wavelengths is:

$\lambda(m_1) : \lambda(m_2) = 3 : 4$

Answer 3

Given: 
$m_1 = 100 \, \text{g}$
$V_1 = V_1 \, \text{m/s}$
$m_2 = 50 \, \text{g}$
$V_2 = 1.5 V_1 \, \text{m/s}$

Now, using the formula for de-Broglie wavelength:
$\lambda = \frac{h}{mv}$

$\Rightarrow \frac{\lambda_1}{\lambda_2} = \frac{m_2 V_2}{m_1 V_1} = \frac{50 \times 1.5 V_1}{100 V_1} = \frac{3}{4}$

So, the correct option is (B) : 3 : 4