The Problem:

A balloon is rising from rest with a constant acceleration of 9/8. At a certain height h, a stone is released from the balloon. We need to find the time taken by the stone to reach the ground.

Solution:

To solve this problem, we can break it down into two parts: the motion of the balloon and the motion of the stone.

Motion of the Balloon:

Let's consider the motion of the balloon first. We are given that the balloon is rising from rest with a constant acceleration of 9/8. This means that its velocity is increasing at a rate of 9/8 m/s^2.

We can use the kinematic equation v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time taken. Since the balloon starts from rest, its initial velocity u is 0.

Using the equation, we can find the time taken by the balloon to reach height h. Let's denote this time as T.

v = u + at
0 + (9/8)T = T * (9/8)
0 = 0

This shows that the balloon reaches height h in no time, which means that the stone is released instantaneously.

Motion of the Stone:

Once the stone is released from the balloon, it starts falling downwards under the influence of gravity. The stone experiences a constant acceleration of -9.8 m/s^2 (negative because it is in the opposite direction to the motion of the balloon).

We can use the kinematic equation s = ut + (1/2)at^2, where s is the distance traveled, u is the initial velocity, a is the acceleration, and t is the time taken. Since the stone starts from rest, its initial velocity u is 0.

We need to find the time taken by the stone to reach the ground, which is the distance traveled by the stone from height h to the ground.

s = ut + (1/2)at^2
s = 0 + (1/2)(-9.8)t^2
h = -4.9t^2
t^2 = -h/4.9
t = sqrt(-h/4.9)

Conclusion:

The time taken by the stone to reach the ground is sqrt(-h/4.9). However, it's important to note that the value of h cannot be negative. If h is positive, the stone will take the square root of a positive number, resulting in a real positive value for the time taken. If h is zero, the stone will take zero time to reach the ground.

Answer may be nearly equal to 1.4√h seconds.