Table-tennis ball has mass 10g and s peed of 90m/s. if speed can be measured within an accuracy of 4%. What will be the uncertainty in speed and position?

Options

A. `1.46xx10^(-33)`
B. `1.527xx10^(-34)`
C. `1.5xx10^(-34)`
D. None of these

Explanation

To solve the problem regarding the uncertainty in speed and position of a table-tennis ball, we will follow these steps:

### Step 1: Identify the given values
- Mass of the table-tennis ball (m) = 10 g = 10 × 10^(-3) kg = 0.01 kg
- Speed of the ball (v) = 90 m/s
- Accuracy of speed measurement = 4%

### Step 2: Calculate the uncertainty in speed (Δv)
The uncertainty in speed can be calculated using the percentage accuracy:
\[ \Delta v = \text{Accuracy} \times v \]
\[ \Delta v = \frac{4}{100} \times 90 \text{ m/s} = 0.04 \times 90 = 3.6 \text{ m/s} \]

### Step 3: Use Heisenberg's Uncertainty Principle
According to Heisenberg's uncertainty principle:
\[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \]
Where:
- \(\Delta p\) = uncertainty in momentum
- \(h\) = Planck's constant = \(6.626 \times 10^{-34} \text{ Js}\)

The momentum (p) is given by:
\[ p = m \cdot v \]
Thus, the uncertainty in momentum (Δp) can be expressed as:
\[ \Delta p = m \cdot \Delta v \]

### Step 4: Calculate Δp
Substituting the values:
\[ \Delta p = (0.01 \text{ kg}) \cdot (3.6 \text{ m/s}) = 0.000036 \text{ kg m/s} = 3.6 \times 10^{-5} \text{ kg m/s} \]

### Step 5: Calculate the uncertainty in position (Δx)
Rearranging the uncertainty principle formula to solve for Δx:
\[ \Delta x \geq \frac{h}{4\pi \Delta p} \]
Substituting the values:
\[ \Delta x \geq \frac{6.626 \times 10^{-34}}{4 \cdot 3.14 \cdot (3.6 \times 10^{-5})} \]

Calculating the denominator:
\[ 4 \cdot 3.14 \cdot (3.6 \times 10^{-5}) \approx 4.52 \times 10^{-4} \]

Now substituting back:
\[ \Delta x \geq \frac{6.626 \times 10^{-34}}{4.52 \times 10^{-4}} \approx 1.46 \times 10^{-30} \text{ m} \]

### Final Answer
The uncertainty in speed (Δv) is 3.6 m/s, and the uncertainty in position (Δx) is approximately \(1.46 \times 10^{-30} \text{ m}\).

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