A stone is dropeed from a height of10 cm above the top of a window 80 cm high. The time taken by the stone to cross the window is `(g = 9.8 ms^(-2))`

Options

A. `1/7 s`
B. `3/7 s`
C. `2/7 s`
D. `4/7 s`

Answer

To solve the problem of finding the time taken by a stone to cross a window when dropped from a height of 10 cm above the top of the window (which is 80 cm high), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**:
- The stone is dropped from a height of 10 cm above the top of an 80 cm high window.
- Therefore, the total distance the stone falls before it crosses the window is 10 cm + 80 cm = 90 cm.

2. **Convert Units**:
- Convert the distances from centimeters to meters:
- Height above the window = 10 cm = 0.10 m
- Height of the window = 80 cm = 0.80 m
- Total distance fallen = 90 cm = 0.90 m

3. **Calculate Time to Fall 10 cm (T1)**:
- Use the kinematic equation:
\[ s = ut + \frac{1}{2}gt^2 \]
where \(s\) is the distance fallen (0.10 m), \(u\) is the initial velocity (0 m/s), \(g\) is the acceleration due to gravity (9.8 m/s²), and \(t\) is the time taken (T1).
- Plugging in the values:
\[ 0.10 = 0 + \frac{1}{2} \times 9.8 \times T1^2 \]
- Rearranging gives:
\[ T1^2 = \frac{0.10 \times 2}{9.8} \]
\[ T1^2 = \frac{0.20}{9.8} \approx 0.02041 \]
- Taking the square root:
\[ T1 \approx \sqrt{0.02041} \approx 0.1429 \text{ seconds} \approx \frac{1}{7} \text{ seconds} \]

4. **Calculate Time to Fall 90 cm (T)**:
- Using the same kinematic equation for the total distance of 0.90 m:
\[ 0.90 = 0 + \frac{1}{2} \times 9.8 \times T^2 \]
- Rearranging gives:
\[ T^2 = \frac{0.90 \times 2}{9.8} \]
\[ T^2 = \frac{1.80}{9.8} \approx 0.18367 \]
- Taking the square root:
\[ T \approx \sqrt{0.18367} \approx 0.428 \text{ seconds} \approx \frac{3}{7} \text{ seconds} \]

5. **Calculate Time Taken to Cross the Window**:
- The time taken to cross the window is given by:
\[ \text{Time to cross window} = T - T1 \]
- Substituting the values:
\[ \text{Time to cross window} = \frac{3}{7} - \frac{1}{7} = \frac{2}{7} \text{ seconds} \]

### Final Answer:
The time taken by the stone to cross the window is \(\frac{2}{7}\) seconds.