Measure of two quantities along with the precision of respective measuring instrument is `A=2.5ms^(-1)+-0.5ms^(-1)`,B=0.10s+-0.01s`. The value of AB will be

Options

A. `(0.25+-0.08)m`
B. `(0.25+-0.5)m`
C. `(0.25+-0.05)m`
D. `(0.25+-0.135)m`

Explanation

To solve the problem, we need to find the product \( AB \) given the values and uncertainties of \( A \) and \( B \). Here are the steps to calculate \( AB \) along with the uncertainties:

### Step 1: Identify the values and uncertainties
- Given:
- \( A = 2.5 \, \text{m/s} \pm 0.5 \, \text{m/s} \)
- \( B = 0.10 \, \text{s} \pm 0.01 \, \text{s} \)

### Step 2: Calculate the product \( AB \)
- The nominal values are:
\[ AB = A \times B = 2.5 \, \text{m/s} \times 0.10 \, \text{s} = 0.25 \, \text{m}^2/\text{s} \]

### Step 3: Calculate the relative uncertainties
- The relative uncertainty in \( A \) is given by:
\[ \frac{\Delta A}{A} = \frac{0.5 \, \text{m/s}}{2.5 \, \text{m/s}} = 0.2 \]
- The relative uncertainty in \( B \) is given by:
\[ \frac{\Delta B}{B} = \frac{0.01 \, \text{s}}{0.10 \, \text{s}} = 0.1 \]

### Step 4: Combine the relative uncertainties
- The total relative uncertainty in the product \( AB \) is:
\[ \frac{\Delta (AB)}{AB} = \frac{\Delta A}{A} + \frac{\Delta B}{B} = 0.2 + 0.1 = 0.3 \]

### Step 5: Calculate the absolute uncertainty in \( AB \)
- The absolute uncertainty in \( AB \) can be calculated as:
\[ \Delta (AB) = AB \times \frac{\Delta (AB)}{AB} = 0.25 \, \text{m}^2/\text{s} \times 0.3 = 0.075 \, \text{m}^2/\text{s} \]

### Step 6: Write the final result
- Therefore, the value of \( AB \) with its uncertainty is:
\[ AB = 0.25 \, \text{m}^2/\text{s} \pm 0.075 \, \text{m}^2/\text{s} \]

### Step 7: Round the uncertainty
- Rounding \( 0.075 \) to two decimal places gives \( 0.08 \).
- Thus, the final answer is:
\[ AB = 0.25 \, \text{m}^2/\text{s} \pm 0.08 \, \text{m}^2/\text{s} \]

### Final Answer
\[ AB = 0.25 \pm 0.08 \, \text{m}^2/\text{s} \]

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