To solve this problem, we need to analyze the motion of both particles A' and B' after they are dropped from the balloon. Since the balloon is moving upward with a constant acceleration of \( g/2 \), we can use kinematic equations to find the positions of both particles at the specified time. Let's break it down step by step.
Understanding the Motion of the Balloon
The balloon is accelerating upwards at \( g/2 \), where \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)). This means the balloon's upward acceleration is about \( 4.905 \, \text{m/s}^2 \). When a particle is dropped from the balloon, it initially has the same upward velocity as the balloon at the moment of release.
Initial Conditions
Let’s denote:
- Time of drop for A' = 0 seconds
- Time of drop for B' = 2 seconds
At the moment A' is dropped, it has an initial velocity \( v_{A'} = v_0 + (g/2) \cdot 0 = v_0 \), where \( v_0 \) is the initial velocity of the balloon at that moment. For B', when it is dropped 2 seconds later, it will have the same initial velocity as A' at that time, which is \( v_0 + (g/2) \cdot 2 \) because the balloon has been accelerating for 2 seconds.
Calculating the Positions
Now, we will calculate the positions of both particles after 6 seconds from the moment B' is dropped.
Position of A' after 6 seconds
A' has been in free fall for 6 seconds. The position \( s_{A'} \) can be calculated using the equation:
s = ut + \frac{1}{2} a t^2
Here, \( u = v_0 \), \( a = -g \) (since gravity acts downward), and \( t = 6 \) seconds:
s_{A'} = v_0 \cdot 6 - \frac{1}{2} g \cdot (6^2)
s_{A'} = 6v_0 - 108.54
Position of B' after 4 seconds
B' has been in free fall for 4 seconds (since it was dropped 2 seconds after A'). The initial velocity for B' at the moment of drop is:
v_{B'} = v_0 + (g/2) \cdot 2 = v_0 + 9.81
Now, we calculate the position \( s_{B'} \):
s_{B'} = (v_0 + 9.81) \cdot 4 - \frac{1}{2} g \cdot (4^2)
s_{B'} = 4(v_0 + 9.81) - 78.48
s_{B'} = 4v_0 + 39.24 - 78.48
s_{B'} = 4v_0 - 39.24
Finding the Separation Distance
The separation distance \( d \) between A' and B' after 6 seconds from the drop of B' is given by:
d = s_{A'} - s_{B'}
Substituting the expressions we found:
d = (6v_0 - 108.54) - (4v_0 - 39.24)
d = 6v_0 - 108.54 - 4v_0 + 39.24
d = 2v_0 - 69.30
Final Result
The separation distance between particles A' and B' after 6 seconds from the drop of B' is:
d = 2v_0 - 69.30
This equation shows that the separation depends on the initial velocity of the balloon at the moment A' was dropped. If you know the value of \( v_0 \), you can calculate the exact distance. If \( v_0 \) is not provided, this is the most simplified form of the answer.
