The uncertainty in the position of a buller weight `20 g` is `+- 10^(-4) m ` .Calculate the uncertainty in its velocity

Explanation

To solve the problem of calculating the uncertainty in the velocity of a bowler ball given its mass and the uncertainty in its position, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given values:**
- Mass of the bowler ball, \( m = 20 \, \text{g} = 20 \times 10^{-3} \, \text{kg} = 0.02 \, \text{kg} \)
- Uncertainty in position, \( \Delta x = \pm 10^{-4} \, \text{m} \)
- Planck's constant, \( h = 6.63 \times 10^{-34} \, \text{J s} \)
- Value of \( \pi \approx 3.14 \)

2. **Use the uncertainty principle:**
The uncertainty principle states that:
\[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \]
where \( \Delta p \) is the uncertainty in momentum.

3. **Relate momentum to velocity:**
The momentum \( p \) is given by:
\[ p = mv \]
Therefore, the uncertainty in momentum can be expressed as:
\[ \Delta p = m \Delta v \]
where \( \Delta v \) is the uncertainty in velocity.

4. **Substitute into the uncertainty principle:**
Replacing \( \Delta p \) in the uncertainty principle:
\[ \Delta x \cdot (m \Delta v) \geq \frac{h}{4\pi} \]

5. **Rearranging for \( \Delta v \):**
\[ \Delta v \geq \frac{h}{4\pi m \Delta x} \]

6. **Substituting the known values:**
\[ \Delta v \geq \frac{6.63 \times 10^{-34}}{4 \cdot 3.14 \cdot 0.02 \cdot 10^{-4}} \]

7. **Calculating the denominator:**
- Calculate \( 4 \cdot 3.14 \cdot 0.02 \cdot 10^{-4} \):
\[ 4 \cdot 3.14 \cdot 0.02 = 0.2512 \]
Thus,
\[ 0.2512 \times 10^{-4} = 2.512 \times 10^{-5} \]

8. **Final calculation of \( \Delta v \):**
\[ \Delta v \geq \frac{6.63 \times 10^{-34}}{2.512 \times 10^{-5}} \approx 2.63 \times 10^{-29} \, \text{m/s} \]

### Final Answer:
The uncertainty in the velocity of the bowler ball is approximately:
\[ \Delta v \approx 2.63 \times 10^{-29} \, \text{m/s} \]