The speed of a 200 g golf ball is 5.0 metre per hour. The wavelength of this ball will be of the order of

Options

A. `10^(-10)m`
B. `10^(-20)m`
C. `10^(-30)m`
D. `10^(-4)m`

Answer

To determine the wavelength of a 200 g golf ball moving at a speed of 5.0 meters per hour, we will use the de Broglie wavelength formula:

\[ \lambda = \frac{h}{mv} \]

where:
- \(\lambda\) is the wavelength,
- \(h\) is Planck's constant (\(6.626 \times 10^{-34} \, \text{Js}\)),
- \(m\) is the mass of the object in kilograms,
- \(v\) is the velocity of the object in meters per second.

### Step 1: Convert mass from grams to kilograms
The mass of the golf ball is given as 200 grams. To convert this to kilograms:

\[ m = 200 \, \text{g} = 200 \times 10^{-3} \, \text{kg} = 0.2 \, \text{kg} \]

**Hint:** Remember that 1 kg = 1000 g, so to convert grams to kilograms, divide by 1000.

### Step 2: Convert speed from meters per hour to meters per second
The speed of the golf ball is given as 5.0 meters per hour. To convert this to meters per second:

\[ v = 5.0 \, \text{m/h} = \frac{5.0 \, \text{m}}{3600 \, \text{s}} \approx 0.0013889 \, \text{m/s} \]

**Hint:** There are 3600 seconds in an hour (60 minutes x 60 seconds), so divide the speed in meters by 3600 to convert to meters per second.

### Step 3: Substitute values into the de Broglie wavelength formula
Now that we have \(m\) and \(v\), we can substitute these values into the de Broglie wavelength formula:

\[ \lambda = \frac{6.626 \times 10^{-34} \, \text{Js}}{(0.2 \, \text{kg}) \times (0.0013889 \, \text{m/s})} \]

### Step 4: Calculate the denominator
First, calculate the product of mass and velocity:

\[ mv = 0.2 \, \text{kg} \times 0.0013889 \, \text{m/s} = 0.00027778 \, \text{kg m/s} \]

### Step 5: Calculate the wavelength
Now substitute this value back into the formula:

\[ \lambda = \frac{6.626 \times 10^{-34}}{0.00027778} \approx 2.39 \times 10^{-30} \, \text{m} \]

### Step 6: Determine the order of the wavelength
The calculated wavelength is approximately \(2.39 \times 10^{-30} \, \text{m}\). The order of this wavelength is \(10^{-30}\).

**Final Answer:** The wavelength of the golf ball is of the order of \(10^{-30} \, \text{m}\).