A student measures that distance traversed in free fall of a body, initially at rest in given time. He uses this data to estimated `g`, the acceleration due to gravity. If the maximum percentage error in measurement of the distance and the time are `e_(1)` and `e_(2)`, respectively, the percentage error in the estimation of `g` is

Options

A. `e_(2) - e_(1)`
B. `e_(1) + 2e_(2)`
C. `e_(1) + e_(2)`
D. `e_(1) - 2e_(2)`

Explanation

To solve the problem of estimating the percentage error in the acceleration due to gravity \( g \) based on the measurements of distance and time, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Relationship**:
The distance \( h \) traversed by a body in free fall from rest is given by the equation:
\[ h = \frac{1}{2} g t^2 \]
From this equation, we can express \( g \) as:
\[ g = \frac{2h}{t^2} \]

2. **Identifying Errors**:
Let \( e_1 \) be the maximum percentage error in the measurement of distance \( h \), and \( e_2 \) be the maximum percentage error in the measurement of time \( t \).

3. **Calculating Percentage Error in \( g \)**:
To find the percentage error in \( g \), we need to differentiate the equation for \( g \):
\[ g = \frac{2h}{t^2} \]
Using the rules of differentiation for products and quotients, we can find the relative error in \( g \):
\[ \frac{\Delta g}{g} = \frac{\Delta h}{h} + 2 \frac{\Delta t}{t} \]
Here, \( \Delta h \) and \( \Delta t \) represent the absolute errors in \( h \) and \( t \) respectively.

4. **Expressing in Terms of Percentage Errors**:
The percentage errors can be expressed as:
\[ \frac{\Delta h}{h} = \frac{e_1}{100} \quad \text{and} \quad \frac{\Delta t}{t} = \frac{e_2}{100} \]
Substituting these into the equation gives:
\[ \frac{\Delta g}{g} = \frac{e_1}{100} + 2 \cdot \frac{e_2}{100} \]

5. **Final Expression for Percentage Error in \( g \)**:
Multiplying the entire equation by 100 to convert to percentage gives:
\[ \text{Percentage error in } g = e_1 + 2e_2 \]

### Conclusion:
Thus, the percentage error in the estimation of \( g \) is given by:
\[ \text{Percentage error in } g = e_1 + 2e_2 \]