What will be the uncertainty in velocity of a bullet with a mass of 10 g whose position is known with `pm0.01mm`?

Options

A. `5.275xx10^(-33)ms^(-1)`
B. `5.275xx10^(-25)ms^(-1)`
C. `5.275xx10^(-5)ms^(-1)`
D. `5.275xx10^(-28)ms^(-1)`

Explanation

To find the uncertainty in the velocity of a bullet with a mass of 10 g and a known position uncertainty of ±0.01 mm, we can use the Heisenberg Uncertainty Principle, which states that:

\[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \]

Where:
- \(\Delta x\) is the uncertainty in position,
- \(\Delta p\) is the uncertainty in momentum,
- \(h\) is Planck's constant (\(6.626 \times 10^{-34} \, \text{Js}\)).

### Step 1: Convert the mass of the bullet to kilograms
The mass of the bullet is given as 10 g. We need to convert this to kilograms since the SI unit of mass is kg.

\[ \text{mass} = 10 \, \text{g} = 0.01 \, \text{kg} \]

**Hint:** Remember to convert grams to kilograms by dividing by 1000.

### Step 2: Convert the position uncertainty to meters
The position uncertainty is given as ±0.01 mm. We need to convert this to meters.

\[ \Delta x = 0.01 \, \text{mm} = 0.01 \times 10^{-3} \, \text{m} = 1 \times 10^{-5} \, \text{m} \]

**Hint:** To convert millimeters to meters, multiply by \(10^{-3}\).

### Step 3: Calculate the uncertainty in momentum (\(\Delta p\))
Using the Heisenberg Uncertainty Principle formula:

\[ \Delta p \geq \frac{h}{4\pi \Delta x} \]

Substituting the known values:

\[ \Delta p \geq \frac{6.626 \times 10^{-34}}{4\pi \times 1 \times 10^{-5}} \]

Calculating the denominator:

\[ 4\pi \times 1 \times 10^{-5} \approx 1.25664 \times 10^{-4} \]

Now substituting this back into the equation:

\[ \Delta p \geq \frac{6.626 \times 10^{-34}}{1.25664 \times 10^{-4}} \approx 5.28 \times 10^{-30} \, \text{kg m/s} \]

**Hint:** Make sure to calculate the denominator accurately to find \(\Delta p\).

### Step 4: Calculate the uncertainty in velocity (\(\Delta v\))
The momentum \(p\) is related to velocity \(v\) by the equation:

\[ p = mv \]

Thus, the uncertainty in momentum can be expressed in terms of uncertainty in velocity:

\[ \Delta p = m \Delta v \]

Rearranging gives us:

\[ \Delta v = \frac{\Delta p}{m} \]

Substituting the values we have:

\[ \Delta v = \frac{5.28 \times 10^{-30}}{0.01} = 5.28 \times 10^{-28} \, \text{m/s} \]

**Hint:** Remember to divide the uncertainty in momentum by the mass to find the uncertainty in velocity.

### Final Answer
The uncertainty in the velocity of the bullet is approximately:

\[ \Delta v \approx 5.28 \times 10^{-28} \, \text{m/s} \]