A steel ball of mass m is moving with a kinetic energy K. The de-Broglie wavelength associated with the ball is

Options

A. `(h)/(2mK)`
B. `sqrt((h)/(2mK))`
C. `(h)/(sqrt(2mK))`
D. none of these

Explanation

To find the de-Broglie wavelength associated with a steel ball of mass \( m \) moving with a kinetic energy \( K \), we can follow these steps:

### Step 1: Understand the de-Broglie Wavelength Formula
The de-Broglie wavelength \( \lambda \) is given by the formula:
\[ \lambda = \frac{h}{p} \]
where \( h \) is Planck's constant and \( p \) is the momentum of the object.

### Step 2: Relate Kinetic Energy to Momentum
The kinetic energy \( K \) of the ball can be expressed in terms of its mass \( m \) and velocity \( v \) as:
\[ K = \frac{1}{2} mv^2 \]
We also know that momentum \( p \) is given by:
\[ p = mv \]

### Step 3: Express Velocity in Terms of Kinetic Energy
From the kinetic energy equation, we can express \( v \) in terms of \( K \):
\[ v = \sqrt{\frac{2K}{m}} \]

### Step 4: Substitute Velocity into the Momentum Equation
Now substituting \( v \) into the momentum equation:
\[ p = m \cdot v = m \cdot \sqrt{\frac{2K}{m}} = \sqrt{2mK} \]

### Step 5: Substitute Momentum into the de-Broglie Wavelength Formula
Now substitute the expression for momentum \( p \) back into the de-Broglie wavelength formula:
\[ \lambda = \frac{h}{p} = \frac{h}{\sqrt{2mK}} \]

### Final Result
Thus, the de-Broglie wavelength associated with the steel ball is:
\[ \lambda = \frac{h}{\sqrt{2mK}} \]