Particle having a mass of 1.0 mg has a velocity of 3600 km/h. Calculate the wavelength of the particle.

Options

A. `6.626xx10^(-31)m`
B. `6.626xx10^(-30)m`
C. `6.626xx10^(-29)m`
D. `6.626xx10^(-28)m`

Answer

To calculate the wavelength of a particle with a mass of 1.0 mg and a velocity of 3600 km/h, we can follow these steps:

### Step 1: Convert Mass from mg to kg
The mass of the particle is given as 1.0 mg. We need to convert this to kilograms.

\[ 1 \text{ mg} = 1 \times 10^{-3} \text{ g} \]
\[ 1 \text{ g} = 1 \times 10^{-3} \text{ kg} \]
Thus,
\[ 1 \text{ mg} = 1 \times 10^{-3} \text{ g} \times 1 \times 10^{-3} \text{ kg/g} = 1 \times 10^{-6} \text{ kg} \]

### Step 2: Convert Velocity from km/h to m/s
The velocity is given as 3600 km/h. We need to convert this to meters per second.

To convert km/h to m/s, we use the conversion factor \( \frac{5}{18} \):
\[ 3600 \text{ km/h} = 3600 \times \frac{5}{18} \text{ m/s} \]
Calculating this gives:
\[ 3600 \times \frac{5}{18} = 1000 \text{ m/s} \]

### Step 3: Calculate Momentum
Momentum \( p \) is given by the formula:
\[ p = mv \]
Substituting the values we have:
\[ p = (1 \times 10^{-6} \text{ kg}) \times (1000 \text{ m/s}) = 1 \times 10^{-3} \text{ kg m/s} \]

### Step 4: Use the Wavelength Formula
The wavelength \( \lambda \) can be calculated using the formula:
\[ \lambda = \frac{h}{p} \]
Where \( h \) is Planck's constant, approximately \( 6.626 \times 10^{-34} \text{ J s} \).

Substituting the values:
\[ \lambda = \frac{6.626 \times 10^{-34} \text{ J s}}{1 \times 10^{-3} \text{ kg m/s}} \]
Calculating this gives:
\[ \lambda = 6.626 \times 10^{-34 + 3} \text{ m} = 6.626 \times 10^{-31} \text{ m} \]

### Final Answer
The wavelength of the particle is:
\[ \lambda = 6.626 \times 10^{-31} \text{ m} \]