In an experiment to measure the height of a bridge by dropping stone into water underneath, if the error in measurement of time of 0.1 s at the end of 2s, then the error in estimation of height of bridge will be

Options

A. 0.49 m
B. 0.98 m
C. 1.96 m
D. 2.12 m

Explanation

To solve the problem of estimating the error in the height of the bridge based on the error in the measurement of time, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We need to find the error in the height of the bridge (Δh) based on the error in the time measurement (Δt) which is given as 0.1 seconds at the end of 2 seconds.

2. **Identify the Relevant Formula**: The height (h) from which the stone is dropped can be calculated using the formula for distance under constant acceleration:
\[ h = ut + \frac{1}{2} a t^2 \]
where:
- \( u \) is the initial velocity (0 m/s, since the stone is dropped),
- \( a \) is the acceleration due to gravity (approximately 9.8 m/s²),
- \( t \) is the time (2 seconds).

3. **Calculate the Height (h)**:
Since the initial velocity \( u = 0 \):
\[ h = 0 \cdot t + \frac{1}{2} \cdot 9.8 \cdot (2^2) \]
\[ h = \frac{1}{2} \cdot 9.8 \cdot 4 = 19.6 \text{ meters} \]

4. **Calculate the Relative Error in Height (Δh)**:
The relative error in height can be calculated using the formula:
\[ \frac{\Delta h}{h} = 2 \cdot \frac{\Delta t}{t} \]
where:
- \( \Delta t = 0.1 \) seconds,
- \( t = 2 \) seconds.

Plugging in the values:
\[ \frac{\Delta h}{19.6} = 2 \cdot \frac{0.1}{2} \]
\[ \frac{\Delta h}{19.6} = \frac{0.1}{1} = 0.1 \]

5. **Calculate the Absolute Error (Δh)**:
Now we can find Δh:
\[ \Delta h = 0.1 \cdot h = 0.1 \cdot 19.6 = 1.96 \text{ meters} \]

6. **Conclusion**: The error in the estimation of the height of the bridge is approximately **1.96 meters**.

### Final Answer:
The error in estimation of the height of the bridge will be **1.96 meters**.

---