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

Explanation

To solve the problem, we need to calculate the product of two quantities \( A \) and \( B \) along with their uncertainties. Here are the steps to find the value of \( AB \) and its associated error:

### Step 1: Identify the given values and uncertainties
- \( A = 2.5 \, \text{ms}^{-1} \) with an uncertainty \( \Delta A = 0.5 \, \text{ms}^{-1} \)
- \( B = 0.10 \, \text{s} \) with an uncertainty \( \Delta B = 0.01 \, \text{s} \)

### Step 2: Calculate the product \( AB \)
To find the product \( AB \):
\[ AB = A \times B = 2.5 \, \text{ms}^{-1} \times 0.10 \, \text{s} = 0.25 \, \text{m} \]

### Step 3: Calculate the relative uncertainties
The relative uncertainty in the product \( AB \) can be calculated using the formula:
\[ \frac{\Delta (AB)}{AB} = \frac{\Delta A}{A} + \frac{\Delta B}{B} \]

Substituting the values:
- Relative uncertainty in \( A \):
\[ \frac{\Delta A}{A} = \frac{0.5}{2.5} = 0.2 \]
- Relative uncertainty in \( B \):
\[ \frac{\Delta B}{B} = \frac{0.01}{0.10} = 0.1 \]

### Step 4: Combine the relative uncertainties
Now, we can add the relative uncertainties:
\[ \frac{\Delta (AB)}{AB} = 0.2 + 0.1 = 0.3 \]

### Step 5: Calculate the absolute uncertainty in \( AB \)
Now, to find the absolute uncertainty \( \Delta (AB) \):
\[ \Delta (AB) = AB \times \frac{\Delta (AB)}{AB} = 0.25 \times 0.3 = 0.075 \, \text{m} \]

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

### Step 7: Rounding the uncertainty
Rounding the uncertainty to two significant figures, we get:
\[ AB = 0.25 \, \text{m} \pm 0.08 \, \text{m} \]

### Final Answer
The final result is:
\[ AB = 0.25 \, \text{m} \pm 0.08 \, \text{m} \]

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