To solve the problem, we need to find the separation between the stone and the aeroplane after 2 seconds of the stone being released. The aeroplane is rising with an upward acceleration of \(5 \, \text{m/s}^2\), and the acceleration due to gravity \(g\) is \(10 \, \text{m/s}^2\).
### Step-by-Step Solution:
1. **Identify the accelerations**:
- The aeroplane has an upward acceleration \(a = 5 \, \text{m/s}^2\).
- The acceleration due to gravity acting downward is \(g = 10 \, \text{m/s}^2\).
2. **Determine the net acceleration of the stone**:
- When the stone is released, it will experience the acceleration due to gravity downward. However, since the aeroplane is moving upward, we need to consider the effective acceleration acting on the stone.
- The effective acceleration acting on the stone after release is:
\[
a_{\text{effective}} = g + a = 10 \, \text{m/s}^2 + 5 \, \text{m/s}^2 = 15 \, \text{m/s}^2
\]
3. **Use the equation of motion**:
- The distance \(S\) traveled by the stone in time \(t\) can be calculated using the equation:
\[
S = ut + \frac{1}{2} a t^2
\]
- Here, the initial velocity \(u = 0\) (since the stone is released), and \(t = 2 \, \text{s}\).
- Substituting the values, we have:
\[
S = 0 + \frac{1}{2} \cdot 15 \cdot (2)^2
\]
- Simplifying further:
\[
S = \frac{1}{2} \cdot 15 \cdot 4 = \frac{1}{2} \cdot 60 = 30 \, \text{m}
\]
4. **Calculate the distance traveled by the aeroplane in 2 seconds**:
- The distance \(S_a\) traveled by the aeroplane in 2 seconds can also be calculated using the same equation of motion:
\[
S_a = u_a t + \frac{1}{2} a_a t^2
\]
- Here, the initial velocity of the aeroplane \(u_a\) is \(0\) (assuming it starts from rest), and the upward acceleration \(a_a = 5 \, \text{m/s}^2\):
\[
S_a = 0 + \frac{1}{2} \cdot 5 \cdot (2)^2 = \frac{1}{2} \cdot 5 \cdot 4 = 10 \, \text{m}
\]
5. **Calculate the separation between the stone and the aeroplane**:
- The separation \(D\) between the stone and the aeroplane after 2 seconds is given by:
\[
D = S - S_a = 30 \, \text{m} - 10 \, \text{m} = 20 \, \text{m}
\]
### Final Answer:
The separation between the stone and the aeroplane after 2 seconds is \(20 \, \text{m}\).
