The de Broglie wavelength associated with a ball of mass 200 g and moving at a speed of 5 metres/hour, is of the order of (`h = 6.625xx10^(-34)`J s) is

Options

A. (a) `10^(-15)`m
B. (b) `10^(-20)`m
C. (c) `10^(-25)`m
D. (d) `10^(-30)`m

Explanation

To find the de Broglie wavelength associated with a ball of mass 200 g moving at a speed of 5 meters/hour, we can follow these steps:

### Step 1: Understand the formula for de Broglie wavelength
The de Broglie wavelength (λ) 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: Express momentum in terms of mass and velocity
Momentum (p) can be expressed as:
\[ p = m \cdot v \]
where \(m\) is the mass and \(v\) is the velocity of the object.

### Step 3: Substitute momentum into the de Broglie wavelength formula
Substituting the expression for momentum into the de Broglie wavelength formula, we get:
\[ \lambda = \frac{h}{m \cdot v} \]

### Step 4: Convert mass from grams to kilograms
The mass of the ball is given as 200 g. To convert this to kilograms:
\[ m = \frac{200 \, \text{g}}{1000} = 0.2 \, \text{kg} \]

### Step 5: Convert speed from meters per hour to meters per second
The speed is given as 5 meters/hour. To convert this to meters per second:
\[ v = \frac{5 \, \text{m}}{1 \, \text{hour}} = \frac{5 \, \text{m}}{3600 \, \text{seconds}} \approx 0.00138889 \, \text{m/s} \]

### Step 6: Substitute values into the de Broglie wavelength formula
Now, substituting the values of \(h\), \(m\), and \(v\) into the formula:
\[ \lambda = \frac{6.625 \times 10^{-34} \, \text{J s}}{0.2 \, \text{kg} \cdot 0.00138889 \, \text{m/s}} \]

### Step 7: Calculate the denominator
Calculating the denominator:
\[ 0.2 \, \text{kg} \cdot 0.00138889 \, \text{m/s} = 0.000277778 \, \text{kg m/s} \]

### Step 8: Calculate the de Broglie wavelength
Now, substituting this value back into the equation:
\[ \lambda = \frac{6.625 \times 10^{-34}}{0.000277778} \approx 2.386 \times 10^{-30} \, \text{m} \]

### Step 9: Determine the order of the wavelength
The order of the wavelength is approximately \(10^{-30} \, \text{m}\).

### Final Answer
Thus, the de Broglie wavelength associated with the ball is of the order of:
\[ \lambda \approx 10^{-30} \, \text{m} \]