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 solve the problem of calculating the wavelength of a particle with a mass of 1.0 mg and a velocity of 3600 km/h, we will use the de Broglie wavelength formula. Here’s a step-by-step solution:

### Step 1: Convert Mass to Kilograms
The mass of the particle is given as 1.0 mg (milligrams). We need to convert this to kilograms (kg) since the SI unit for mass is kg.

\[ \text{Mass} = 1.0 \, \text{mg} = 1.0 \times 10^{-3} \, \text{g} = 1.0 \times 10^{-6} \, \text{kg} \]

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

\[ \text{Velocity} = 3600 \, \text{km/h} = 3600 \times \frac{1000 \, \text{m}}{1 \, \text{km}} \times \frac{1 \, \text{h}}{3600 \, \text{s}} = 1000 \, \text{m/s} \]

### Step 3: Calculate Momentum
Momentum (p) is calculated using the formula:

\[ p = mv \]

Substituting the values of mass and velocity:

\[ p = (1.0 \times 10^{-6} \, \text{kg}) \times (1000 \, \text{m/s}) = 1.0 \times 10^{-3} \, \text{kg m/s} \]

### Step 4: Use the de Broglie Wavelength Formula
The de Broglie wavelength (λ) is given by the formula:

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

Where \( h \) is Planck's constant, approximately \( 6.626 \times 10^{-34} \, \text{Js} \).

Substituting the values:

\[ \lambda = \frac{6.626 \times 10^{-34} \, \text{Js}}{1.0 \times 10^{-3} \, \text{kg m/s}} = 6.626 \times 10^{-31} \, \text{m} \]

### Final Answer
The wavelength of the particle is:

\[ \lambda = 6.626 \times 10^{-31} \, \text{m} \]

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