A stone is dropped from a hot air balloon that is rising at a constant speed of 2.2 m/s. If the stone takes 4.8 seconds to reach the ground then …
a) What was the height of the balloon at the moment the stone was released?
b) What is the height of the balloon at the time the stone hit the ground?
hf=0=hi+vi*t-1/2 g t^2
you are given vi, and time t.
solve for initial height hi
hf=hi+2.2*4.8 use hi form above, solve for hf the final height of the balloon at time of stone crash.
To answer these questions, we can use the equations of motion for an object in free fall. The key information we have is that the stone takes 4.8 seconds to reach the ground and that the balloon is rising at a constant speed of 2.2 m/s.
a) To find the height of the balloon at the moment the stone was released, we need to determine the time it takes for the stone to reach the ground from the moment it was released. Since the stone falls under the influence of gravity while the balloon is rising, we can equate the distance traveled by the stone to the distance traveled by the balloon. Let's represent the height of the balloon at the moment the stone was released as h1.
The distance covered by the stone as it falls is given by the equation:
distance = (1/2) * acceleration * time^2
Since the stone is in free fall near the surface of the Earth, the acceleration due to gravity is approximately 9.8 m/s^2. Therefore, the distance traveled by the stone is:
distance_stone = (1/2) * 9.8 * (4.8)^2
The distance covered by the balloon as it rises is:
distance_balloon = speed * time
Since the stone takes 4.8 seconds to fall, the distance traveled by the balloon is:
distance_balloon = 2.2 * 4.8
Since the distance traveled by the stone and the balloon are equal, we can set up the equation:
(1/2) * 9.8 * (4.8)^2 = 2.2 * 4.8 + h1
Simplifying and solving for h1:
h1 = (1/2) * 9.8 * (4.8)^2 - 2.2 * 4.8
Therefore, the height of the balloon at the moment the stone was released is h1.
b) To find the height of the balloon at the time the stone hit the ground, we need to find the total distance traveled by the balloon from the moment the stone was released until the stone hits the ground. Let's represent the height of the balloon at the time the stone hits the ground as h2.
The distance covered by the balloon is given by:
distance_balloon = speed * time
Since the stone takes 4.8 seconds to fall, the total time the balloon rises while the stone is falling is 4.8 seconds. Therefore, the distance covered by the balloon is:
distance_balloon = 2.2 * 4.8
Since the height of the balloon at the time the stone was released was h1, the total height of the balloon at the time the stone hits the ground is:
h2 = h1 + distance_balloon
Substituting the value of h1 and distance_balloon from part a, we can find the height of the balloon at the time the stone hits the ground.
I hope this explanation helps you understand how to solve the problem.