To solve the problem step by step, we will break it down into parts (a) and (b) as required.
### Given Data:
- Height of the elevator (h) = 2.7 m
- Acceleration of the elevator (a) = 1.2 m/s²
- Time before the bolt falls (t₀) = 2 s
- Acceleration due to gravity (g) = 9.8 m/s²
### Part (a): Time after which the bolt hits the floor of the elevator
1. **Calculate the velocity of the elevator after 2 seconds**:
\[
v_{\text{elevator}} = u + at = 0 + (1.2 \, \text{m/s}^2)(2 \, \text{s}) = 2.4 \, \text{m/s}
\]
**Hint**: Remember that the initial velocity \(u\) of the elevator is 0 since it starts from rest.
2. **Determine the effective acceleration of the bolt with respect to the elevator**:
The bolt is falling under gravity while the elevator is accelerating upwards. Therefore, the effective acceleration \(a_{\text{effective}}\) of the bolt with respect to the elevator is:
\[
a_{\text{effective}} = g - a = 9.8 \, \text{m/s}^2 - 1.2 \, \text{m/s}^2 = 8.6 \, \text{m/s}^2
\]
3. **Use the equation of motion to find the time \(t\) it takes for the bolt to hit the floor of the elevator**:
The distance \(s\) the bolt falls is equal to the height of the elevator:
\[
s = ut + \frac{1}{2} a_{\text{effective}} t^2
\]
Here, \(u = 2.4 \, \text{m/s}\) (initial velocity of the bolt when it starts falling) and \(s = 2.7 \, \text{m}\):
\[
2.7 = (2.4)t + \frac{1}{2}(8.6)t^2
\]
Rearranging gives:
\[
4.3t^2 + 2.4t - 2.7 = 0
\]
4. **Solve the quadratic equation**:
Using the quadratic formula \(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
- \(a = 4.3\)
- \(b = 2.4\)
- \(c = -2.7\)
\[
t = \frac{-2.4 \pm \sqrt{(2.4)^2 - 4 \cdot 4.3 \cdot (-2.7)}}{2 \cdot 4.3}
\]
Calculate the discriminant:
\[
(2.4)^2 + 4 \cdot 4.3 \cdot 2.7 = 5.76 + 46.44 = 52.2
\]
Now calculate \(t\):
\[
t = \frac{-2.4 \pm \sqrt{52.2}}{8.6}
\]
Taking the positive root:
\[
t \approx \frac{-2.4 + 7.22}{8.6} \approx \frac{4.82}{8.6} \approx 0.56 \, \text{s}
\]
### Part (b): Net displacement and distance travelled by the bolt with respect to the Earth
1. **Calculate the total time from the start to when the bolt hits the floor**:
Total time \(T\) from the start of the elevator to when the bolt hits the floor:
\[
T = t₀ + t = 2 \, \text{s} + 0.56 \, \text{s} = 2.56 \, \text{s}
\]
2. **Calculate the displacement of the bolt with respect to the Earth**:
The displacement \(S\) can be calculated using:
\[
S = ut + \frac{1}{2} g t^2
\]
Here, \(u = 2.4 \, \text{m/s}\) and \(g = 9.8 \, \text{m/s}^2\):
\[
S = (2.4)(2.56) + \frac{1}{2}(9.8)(2.56)^2
\]
Calculate \(S\):
\[
S \approx 6.144 + \frac{1}{2}(9.8)(6.5536) \approx 6.144 + 32.16 \approx 38.304 \, \text{m}
\]
3. **Calculate the distance travelled by the bolt**:
The distance travelled by the bolt is the sum of the distance it falls and the distance the elevator moves upwards:
- Distance fallen by the bolt = height of the elevator = 2.7 m
- Distance moved by the elevator in 2.56 seconds:
\[
d_{\text{elevator}} = \frac{1}{2} a t^2 = \frac{1}{2}(1.2)(2.56)^2 \approx 3.7 \, \text{m}
\]
Total distance travelled by the bolt:
\[
d_{\text{total}} = d_{\text{elevator}} + d_{\text{bolt}} = 3.7 + 2.7 \approx 6.4 \, \text{m}
\]
### Final Answers:
(a) The time after which the bolt hits the floor of the elevator is approximately **0.56 seconds**.
(b) The net displacement of the bolt with respect to the Earth is approximately **38.304 meters**, and the total distance travelled by the bolt is approximately **6.4 meters**.
