If kinetic energy of a proton is increased nine times, the wavelength of the de - Broglie wave associated with it would become :-

Options

A. 3 times
B. 9 times
C. `1/3` times
D. `1/9` times

Answer

To solve the problem of how the de Broglie wavelength changes when the kinetic energy of a proton is increased nine times, we can follow these steps:

### Step 1: Understand the relationship between kinetic energy and de Broglie wavelength
The kinetic energy (KE) of a particle is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
where \( m \) is the mass and \( v \) is the velocity of the proton.

The de Broglie wavelength (\( \lambda \)) is given by:
\[ \lambda = \frac{h}{mv} \]
where \( h \) is Planck's constant.

### Step 2: Express velocity in terms of kinetic energy
From the kinetic energy formula, we can express the velocity \( v \) as:
\[ v = \sqrt{\frac{2 \cdot KE}{m}} \]

### Step 3: Substitute velocity into the de Broglie wavelength equation
Substituting the expression for \( v \) into the de Broglie wavelength formula:
\[ \lambda = \frac{h}{m \cdot \sqrt{\frac{2 \cdot KE}{m}}} \]
This simplifies to:
\[ \lambda = \frac{h}{\sqrt{2m \cdot KE}} \]

### Step 4: Determine the relationship between initial and final kinetic energy
Let the initial kinetic energy be \( KE_1 \) and the final kinetic energy be \( KE_2 \). According to the problem, \( KE_2 = 9 \cdot KE_1 \).

### Step 5: Find the ratio of the wavelengths
Using the relationship derived, we can express the wavelengths:
\[ \lambda_1 = \frac{h}{\sqrt{2m \cdot KE_1}} \quad \text{and} \quad \lambda_2 = \frac{h}{\sqrt{2m \cdot KE_2}} \]
Now substituting \( KE_2 \):
\[ \lambda_2 = \frac{h}{\sqrt{2m \cdot (9 \cdot KE_1)}} = \frac{h}{\sqrt{9 \cdot 2m \cdot KE_1}} = \frac{h}{3\sqrt{2m \cdot KE_1}} = \frac{1}{3} \lambda_1 \]

### Step 6: Conclusion
Thus, the final de Broglie wavelength \( \lambda_2 \) is:
\[ \lambda_2 = \frac{1}{3} \lambda_1 \]

### Final Answer
The wavelength of the de Broglie wave associated with the proton would become \( \frac{1}{3} \) times the initial wavelength.

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