A ball weighing 10 g is moving with a velocity of `90ms^(-1)`. If the uncertainty in its velocity is 5%, then the uncertainty in its position is _______ `xx10^(-33)m`. Rounded off to the nearest integer)
[Given `h=6.63xx10^(-34)Js` ]

Answer

To solve the problem, we will use the Heisenberg Uncertainty Principle, which states that the product of the uncertainties in position (Δx) and momentum (Δp) is at least a constant value given by:

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

Where:
- \( h \) is Planck's constant, \( h = 6.626 \times 10^{-34} \, \text{Js} \)
- \( \Delta p \) is the uncertainty in momentum, which can be calculated as \( m \cdot \Delta v \)

### Step 1: Calculate the mass in kg
The mass of the ball is given as 10 g. We need to convert this to kilograms:

\[ m = 10 \, \text{g} = \frac{10}{1000} \, \text{kg} = 0.01 \, \text{kg} \]

### Step 2: Calculate the uncertainty in velocity (Δv)
The velocity of the ball is given as \( 90 \, \text{ms}^{-1} \) and the uncertainty in velocity is 5%. We can calculate Δv as follows:

\[ \Delta v = 0.05 \times 90 \, \text{ms}^{-1} = 4.5 \, \text{ms}^{-1} \]

### Step 3: Calculate the uncertainty in momentum (Δp)
Now, we can calculate the uncertainty in momentum using the formula \( \Delta p = m \cdot \Delta v \):

\[ \Delta p = 0.01 \, \text{kg} \cdot 4.5 \, \text{ms}^{-1} = 0.000045 \, \text{kg m/s} = 4.5 \times 10^{-5} \, \text{kg m/s} \]

### Step 4: Use the Heisenberg Uncertainty Principle
Now we can substitute Δp into the Heisenberg Uncertainty Principle equation:

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

Rearranging gives us:

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

### Step 5: Substitute values into the equation
Substituting the known values:

\[ \Delta x \geq \frac{6.626 \times 10^{-34}}{4 \cdot \frac{22}{7} \cdot 4.5 \times 10^{-5}} \]

Calculating the denominator:

\[ 4 \cdot \frac{22}{7} \cdot 4.5 \times 10^{-5} \approx 4 \cdot 3.14 \cdot 4.5 \times 10^{-5} \approx 5.654 \times 10^{-4} \]

Now, substituting back into the equation for Δx:

\[ \Delta x \geq \frac{6.626 \times 10^{-34}}{5.654 \times 10^{-4}} \approx 1.17 \times 10^{-30} \, \text{m} \]

### Step 6: Convert to the required format
Since we need the answer in the form \( xx \times 10^{-33} \, \text{m} \):

\[ 1.17 \times 10^{-30} = 11.7 \times 10^{-31} = 1.17 \times 10^{-33} \, \text{m} \]

### Step 7: Round to the nearest integer
Rounding off \( 1.17 \) to the nearest integer gives us \( 1 \).

Thus, the final answer is:

\[ \text{The uncertainty in position is } 1 \times 10^{-33} \, \text{m}. \]