A small bag of sand is released from an ascending hot-air balloon whose upward constant velocity is v0 = 1.35 m/s. Knowing that at the time of the release the balloon was 81.8 m above the ground, determine the time, τ, it takes the bag to reach the ground from the moment of its release.
To solve this problem, we can use the equation of motion for free fall:
h = v0 * t + (1/2) * g * t^2
Where:
h = distance traveled (81.8 m)
v0 = initial velocity (1.35 m/s)
g = acceleration due to gravity (9.8 m/s^2)
t = time
Since the sandbag is released, its initial velocity is zero (v0 = 0 m/s). We can simplify the equation to:
h = (1/2) * g * t^2
Plugging in the given values:
81.8 = (1/2) * 9.8 * t^2
Now we can solve for t. Rearranging the equation:
t^2 = (2 * 81.8) / 9.8
t^2 = 16.72
Taking the square root of both sides:
t = √16.72
t ≈ 4.09 s
Therefore, it takes approximately 4.09 seconds for the bag of sand to reach the ground from the moment it was released.
To determine the time it takes for the bag of sand to reach the ground from the moment of its release, we need to use the kinematic equation that relates distance, initial velocity, acceleration, and time:
d = v0 * t + (1/2) * a * t^2
Where:
d is the distance traveled by the bag of sand (81.8 m),
v0 is the initial velocity of the balloon (1.35 m/s),
t is the time it takes for the bag to reach the ground, and
a is the acceleration (which is equal to the acceleration due to gravity, approximately 9.8 m/s^2).
Since the bag is released from an ascending balloon, its initial velocity is in the upward direction. This means that the acceleration due to gravity acts in the opposite direction, which is downward. So, we need to use a negative value for acceleration.
Plugging the given values into the equation:
81.8 m = (1.35 m/s) * t + (1/2) * (-9.8 m/s^2) * t^2
Simplifying the equation:
81.8 = 1.35t - 4.9t^2
Rearranging the equation to form a quadratic equation:
4.9t^2 - 1.35t + 81.8 = 0
Now we can solve this quadratic equation using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 4.9, b = -1.35, and c = 81.8. Plugging these values into the quadratic formula:
t = (-(-1.35) ± √((-1.35)^2 - 4 * 4.9 * 81.8)) / (2 * 4.9)
Calculating the values inside the square root:
t = (1.35 ± √(1.8225 - 1597.84)) / 9.8
t = (1.35 ± √(-1596.0175)) / 9.8
Since we have a negative value inside the square root, it means that the quadratic equation does not have a real solution. This implies that the bag of sand will not reach the ground and will stay at a height of 81.8 m above the ground.
Therefore, the bag of sand will not reach the ground from the moment of its release.
well, the height of the bag is
h(t) = 81.8 + 1.35t - 4.9t^2
so just solve for t when h=0.