A hot air balloon is traveling vertically upward at a constant speed of 2.9 m/s. When it is 13 m above the ground, a package is released from the balloon. After it is released, for how long is the package in the air? The acceleration of gravity is 9.8 m/s^2.
the height h is
h = 13 + 2.9t - 4.9t^2
so find t when h=0
h = 1/2 g t^2 + v t + h0 ... the height (h) is zero when the package lands
0 = -4.9 t^2 + 2.9 t + 13
solve the quadratic for t , the "flight time"
... you want the positive root
To determine how long the package is in the air after it is released from the hot air balloon, we need to consider the motion of the package. Since the package is only under the influence of gravity once it is released, we can treat its motion as free fall.
First, let's find the time it takes for the package to hit the ground after being released. We can use the equation for displacement in free fall:
h = (1/2) * g * t^2
Where:
h is the initial height of the package (13 m)
g is the acceleration due to gravity (-9.8 m/s^2, negative because it acts downward)
t is the time
Rearranging the equation to solve for time, we have:
t^2 = (2 * h) / g
t = sqrt((2 * h) / g)
Substituting the given values, we have:
t = sqrt((2 * 13) / 9.8)
t = sqrt(26 / 9.8)
t = sqrt(2.653) ā 1.63 s
So it takes approximately 1.63 seconds for the package to hit the ground after being released.
However, we're interested in the total time the package is in the air, including the time it takes for the balloon to rise to a height of 13 m. To find this, we need to determine the time it takes for the balloon to reach that height.
Since the hot air balloon is traveling upward at a constant speed, its motion is uniform. We can use the equation for uniform motion:
v = d / t
Where:
v is the upward velocity of the balloon (2.9 m/s)
d is the distance traveled by the balloon (13 m)
t is the time
Rearranging the equation to solve for time, we have:
t = d / v
t = 13 / 2.9
t ā 4.48 s
Therefore, it takes approximately 4.48 seconds for the balloon to reach a height of 13 m.
Now, to find the total time the package is in the air, we add the time it takes for the balloon to rise and the time it takes for the package to hit the ground:
Total time = time for balloon to reach 13m + time for package to hit the ground
Total time = 4.48 s + 1.63 s
Total time ā 6.11 s
Hence, the package is in the air for approximately 6.11 seconds after being released from the hot air balloon.