A hot-air balloon ascends at the rate of 12m/s when it is 80m above the ground at which time a package is dropped over the side .
(a) how long would the package take to reach the ground ?
(b) with what speed would it hit the ground?
No name: ARe you looking for answers to turn in and don't wnat your teacher to know? That is rather gutless.
hf=hi+vi*t+1/2 g t^2
0=80+12m/s*t-4.9t^2
solve for time t, with the use of the quadratic equation.
Speed:
Initial PE+initial KE= finalKE
mg*80+1/2 m 12^2 = 1/2 m vf^2
solve for vfinal, vf
To find the answers to these questions, we can break the problem into two parts: the time it takes for the package to reach the ground and the speed at which it would hit the ground.
(a) To determine how long the package would take to reach the ground, we need to use the concept of free-fall motion. The package is dropped vertically from the hot-air balloon, so it will experience free-fall acceleration due to gravity.
We can use the equation of motion:
h = ut + (1/2)gt^2
Where:
- h is the height (in meters)
- u is the initial velocity (in this case, 0 m/s because the package is dropped and not thrown)
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
- t is the time (in seconds)
Given that the initial height is 80 m and the initial velocity is 0 m/s, we can rearrange the equation to solve for time:
h = (1/2)gt^2
80 = (1/2)(9.8)t^2
To solve for t, we can divide both sides of the equation by (1/2)(9.8):
t^2 = (80) / ((1/2)(9.8))
Simplifying further:
t^2 = (80) / (4.9)
t^2 = 16.32653
Taking the square root of both sides:
t ≈ 4.04 seconds
Therefore, it would take approximately 4.04 seconds for the package to reach the ground.
(b) To find the speed at which the package hits the ground, we can use the formula for the final velocity in free-fall motion:
v = u + gt
Where:
- v is the final velocity
- u is the initial velocity (0 m/s)
- g is the acceleration due to gravity (9.8 m/s^2)
- t is the time (4.04 seconds as calculated in part a)
Substituting the values into the formula:
v = 0 + (9.8)(4.04)
v = 39.6
Therefore, the package would hit the ground with a speed of approximately 39.6 m/s.