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We are tasked with finding the **relative error** in the determination of time \( t \) for a ball dropped from height \( H \) using the formula:

\[
t = \sqrt{\frac{2H}{g}}
\]

where:
- \( H = 1.490 \, \text{m} \) with a probable error of \( 0.018 \, \text{m} \),
- \( g = 9.82 \, \text{m/s}^2 \) with a probable error of \( 0.44 \, \text{m/s}^2 \).

### Step 1: Expression for the relative error of \( t \)

To find the relative error in \( t \), we first need to differentiate \( t = \sqrt{\frac{2H}{g}} \) with respect to both \( H \) and \( g \), treating the errors as small differentials:

\[
\frac{\Delta t}{t} = \frac{1}{2} \left( \frac{\Delta H}{H} + \frac{\Delta g}{g} \right)
\]

### Step 2: Calculate the relative errors for \( H \) and \( g \)

- Relative error in height \( H \):
\[
\frac{\Delta H}{H} = \frac{0.018}{1.490}
\]

- Relative error in gravitational acceleration \( g \):
\[
\frac{\Delta g}{g} = \frac{0.44}{9.82}
\]

### Step 3: Combine the relative errors

Using the formula for the relative error in \( t \), we substitute the relative errors from above:

\[
\frac{\Delta t}{t} = \frac{1}{2} \left( \frac{0.018}{1.490} + \frac{0.44}{9.82} \right)
\]

Now, let me calculate the final result.The relative errors for height \( H \) and gravitational acceleration \( g \) are:

- Relative error in \( H \): \( 0.0121 \) (or 1.21%)
- Relative error in \( g \): \( 0.0448 \) (or 4.48%)

Thus, the relative error in the time \( t \) is approximately \( 0.0284 \), or 2.84%.

Would you like more details on how to propagate errors, or do you have any questions about the steps?

Here are some related questions:
1. How would the time \( t \) change if the height \( H \) was doubled?
2. How would the relative error in time change if the error in \( H \) was reduced by half?
3. How does the uncertainty in \( g \) affect the overall error more than the uncertainty in \( H \)?
4. What assumptions are made when calculating relative errors for small differentials?
5. How would you account for systematic errors in a real experiment when measuring \( H \)?

**Tip**: Always express uncertainties with the same precision as the measured value to maintain consistency in reporting data.

If a ball is dropped from a height H with a given probable error in height and gravitational acceleration, what is the relative error in the determination of the time t?

Learn how to calculate the relative error in the time determination when a ball is dropped from a height using error propagation. This problem involves the differentiation of time with respect to height and gravitational acceleration and combines relative errors in measurements. Suitable for college-level students.

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