A hot air balloon is traveling vertically upward at a constant speed of 5.3 m/s. When it is 25 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. Answer in units of s.
Any way someone could help me out with this question? I'm just learning physics. Thanks in advance!
Of course! I'd be happy to help you with this physics question.
To find the time the package is in the air, we need to consider the motion of the package after it is released. Since the hot air balloon is traveling vertically upward at a constant speed, we know that the initial velocity of the package is also 5.3 m/s but directed downward.
First, let's calculate the time it takes for the package to reach the ground after it's released:
We can use the formula for vertical motion under constant acceleration:
h = v0t + (1/2)gt^2
Where:
h is the displacement or the distance above the ground, which is 25 m.
v0 is the initial velocity, which is -5.3 m/s (negative because it's directed downward).
g is the acceleration due to gravity, which is 9.8 m/s^2.
t is the time we're looking for.
Now, plug in the values into the equation:
25 = -5.3t + (1/2)(9.8)t^2
Rearranging the equation, we get:
(1/2)(9.8)t^2 - 5.3t + 25 = 0
This is a quadratic equation, and we can solve it using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / 2a
Substituting the coefficients into the formula:
t = (-(-5.3) ± √((-5.3)^2 - 4(1/2)(9.8)(25))) / 2(1/2)(9.8)
Simplifying further, we get:
t = (5.3 ± √(5.3^2 - 4(4.9)(25))) / (9.8)
Evaluating the expression inside the square root:
t = (5.3 ± √(28.09 - 490)) / (9.8)
t = (5.3 ± √(-461.91)) / (9.8)
Since we cannot take the square root of a negative number, we know that the package does not reach the ground. Therefore, it never takes 25 m to reach the ground.
So, the package is in the air for the entire duration the hot air balloon is traveling vertically upward at a constant speed, which means it is released and remains in the air for the entire duration of the balloon's motion.
Therefore, the answer is that the package is in the air for the same duration as the hot air balloon travels vertically upward at a constant speed, which is the total time until the balloon stops or changes its motion. However, the problem statement does not provide information about how long the balloon will continue to ascend, so we cannot determine the exact time the package is in the air without additional information.