To calculate the time it takes for the object to reach the ground when dropped from a balloon moving at a steady velocity, we need to consider the effects of both the balloon's upward velocity and the downward acceleration due to gravity.
Given: Upward velocity of the balloon (v_balloon) = 20 m/s Height of the balloon above the ground (h_balloon) = 60 m
The object will have an initial velocity of 20 m/s upward, relative to the ground, due to the motion of the balloon. However, once dropped, it will start experiencing the acceleration due to gravity, which is approximately 9.8 m/s² downward on Earth.
To find the time it takes for the object to reach the ground, we can use the following equation of motion:
h = v_initial * t + (1/2) * a * t^2
where: h = height (60 m) v_initial = initial velocity (20 m/s) a = acceleration (-9.8 m/s²) t = time
Substituting the values:
60 = 20 * t + (1/2) * (-9.8) * t^2
Rearranging the equation:
4.9 * t^2 + 20 * t - 60 = 0
This is a quadratic equation. Solving it will give us the time it takes for the object to reach the ground.
Using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
For this equation, a = 4.9, b = 20, and c = -60.
Calculating:
t = (-20 ± √(20^2 - 4 * 4.9 * -60)) / (2 * 4.9)
t ≈ (-20 ± √(400 + 1176)) / 9.8
t ≈ (-20 ± √(1576)) / 9.8
t ≈ (-20 ± 39.7) / 9.8
Since time cannot be negative, we can disregard the negative value:
t ≈ (-20 + 39.7) / 9.8
t ≈ 19.7 / 9.8
t ≈ 2.01 seconds (rounded to two decimal places)
Therefore, the object will take approximately 2.01 seconds to reach the ground when dropped from the balloon.