The time that takes for a small bag of sand to reach the ground after it is released from an ascending hot-air balloon whose upward velocity is 2.25 m/s is 2.94 s.
We can find the time that takes for the bag to reach the ground with the following equation:
[tex] y_{f} = y_{i} + v_{i_{y}}t + \frac{1}{2}gt^{2} [/tex] (1)
Where:
[tex] y_{f} [/tex]: is the final height = 35.8 m
[tex] y_{i} [/tex]: is the initial height = 0
[tex] v_{i_{y}} [/tex]: is the initial velocity = 2.25 m/s
t: is the time =?
g: is the acceleration due to gravity = 9.81 m/s²
When the bag is released, it will move upward until it reaches a maximum height and then begins to fall. We can find the time that takes for the bag to reach the maximum height as follows:
[tex] v_{f} = v_{i} - gt [/tex] (2)
Where:
[tex] v_{f} [/tex]: is the final velocity = 0 (at the maximum height)
[tex] v_{i} [/tex]: is the initial velocity = 2.25 m/s
The time is:
[tex] t = -\frac{v_{f} - v_{i}}{g} = \frac{2.25 m/s}{9.81 m/s^{2}} = 0.23 s [/tex]
This is the rise time it takes for the bag to reach the maximum height, which is equal to the fall time it takes for the bag to reach the same starting point (35.8 m above the ground), so:
[tex] t = 0.23 s*2 = 0.46 s [/tex]
Now, from equation (1) we have:
[tex] 35.8 m = 0 + 2.25 m/s*t + \frac{1}{2}9.81 m/s^{2}*t^{2} [/tex]
By solving the above quadratic equation for t we have:
[tex] t = 2.48 s [/tex]
Hence, the total time is:
[tex] t = 2.48 s + 0.46 s = 2.94 s [/tex]
Therefore, the time that it takes for the bag to reach the ground is 2.94 s.
You can see another example of freefall here: https://brainly.com/question/10555700?referrer=searchResults
I hope it helps you!
