A halogen gas failed balloon released frim the ground,travels vertically up with a constant acceleration of 2m/s^2.Three seconds later,a stone is projected vertically up from the ground to hit the ascending balloon.The stone just manages to touch the balloon at the peak of its path.Find the velocity with which the stone was projected and the hight at which the stone touches the balloon.

Question

A halogen gas failed balloon released frim the ground,travels vertically up with a constant acceleration of 2m/s^2.Three seconds later,a stone is projected vertically up from the ground to hit the ascending balloon.The stone just manages to touch the balloon at the peak of its path.Find the velocity with which the stone was projected and the hight at which the stone touches the balloon.

Answer 1

To solve this problem, we can use the equations of motion for both the balloon and the stone. Let's first consider the balloon's motion.

For the balloon:
Initial velocity, u = 0 (as it was released from rest)
Acceleration, a = 2 m/s^2
Time, t = 3 s (since the stone is projected 3 seconds later)

Using the equation of motion: v = u + at, where v is the final velocity of the balloon, we can calculate v for the balloon.

v = u + at
v = 0 + (2)(3)
v = 6 m/s

So, the velocity of the balloon is 6 m/s when the stone touches it at its peak.

Now, let's consider the motion of the stone.

For the stone:
Initial velocity, u = ?
Acceleration, a = -9.8 m/s^2 (since the stone is going up against the force of gravity)
Final velocity, v = 6 m/s (as the stone touches the balloon at its peak)
Time, t = ?

Using the same equation of motion: v = u + at, we can calculate u and t for the stone.

v = u + at
6 = u + (-9.8)t

We also know that the time it takes for the stone to reach its peak is the same as the time the balloon takes to reach the same point. Therefore, t = 3 s.

Now, we can substitute the values of v and t into the equation and solve for u.

6 = u + (-9.8)(3)
6 = u - 29.4
u = 35.4 m/s

So, the velocity with which the stone was projected is 35.4 m/s.

To calculate the height at which the stone touches the balloon, we need to determine the height reached by the balloon in the 3 seconds it takes for the stone to reach it.

Using the equation of motion: s = ut + (1/2)at^2, where s is the displacement (height), u is the initial velocity, a is the acceleration, and t is the time.

For the balloon:
Initial velocity, u = 0
Acceleration, a = 2 m/s^2
Time, t = 3 s

s = ut + (1/2)at^2
s = 0 + (1/2)(2)(3)^2
s = 9 m

So, the height at which the stone touches the balloon is 9 m.

In summary:
- The stone was projected with a velocity of 35.4 m/s.
- The stone touches the balloon at a height of 9 m.

Answer 2

To solve this problem, we can use the equations of motion for constant acceleration.

Let's start with the motion of the halogen gas failed balloon. It moves vertically up with a constant acceleration of 2 m/s^2. We need to find the velocity with which the stone was projected and the height at which the stone touches the balloon.

Let's analyze the motion of the balloon:

1. We know the acceleration of the balloon is 2 m/s^2, which means the velocity of the balloon changes by 2 m/s every second.
2. After 3 seconds, the balloon has been moving vertically up for 3 seconds with an acceleration of 2 m/s^2. So, at this point, the balloon's velocity is:
velocity = initial velocity + acceleration * time
= 0 + 2 * 3
= 6 m/s

Now, let's analyze the motion of the stone:

1. Since the stone and the balloon intersect at the peak of the balloon's path, we can use the balloon's height at that point as the maximum height reached by the stone.
2. To find the height, we need to determine the time it takes for the stone to reach its maximum height.
3. The stone's initial velocity will be the negative of the balloon's velocity:
initial velocity of the stone = - 6 m/s
4. The stone moves vertically up, so we can use the equation of motion for height:
height = initial velocity * time + (1/2) * acceleration * time^2
5. At the peak of the stone's path, its final velocity will be zero, so:
0 = initial velocity + acceleration * time
In this case:
0 = -6 + (-9.8) * time (taking g = 9.8 m/s^2 as the acceleration due to gravity)
Solving this equation, we find:
time = 6/9.8 ≈ 0.6122 seconds
6. Now, we can substitute the value of time in the equation of motion for height:
height = -6 * 0.6122 + (1/2) * (-9.8) * (0.6122)^2

Calculating this expression will give us the height at which the stone touches the balloon.

So, to summarize:

- The velocity with which the stone was projected is -6 m/s.
- The height at which the stone touches the balloon can be calculated using the equation of motion for height.

I hope this helps! Let me know if you have any further questions.