A hot-air balloon is rising upward with a constant speed of 2.83 m/s. When the balloon is 8.64 m above the ground, the balloonist accidentally drops a compass over the side of the balloon. How much time elapses before the compass hits the ground?
Y = 8.64 + 2.83 t -4.9 t^2 = 0
Solve for t. Take the positive root of the quadratic
t = 1.65
To find the time it takes for the compass to hit the ground, we need to calculate the time based on the vertical distance it falls.
We know that the initial vertical velocity of the compass is 0 m/s because it was dropped from rest.
The acceleration due to gravity (g) is approximately 9.81 m/s², directed downward.
We can use the kinematic equation for motion in the vertical direction:
h = v0t + (1/2)gt²
Where:
h = vertical distance (8.64 m in this case)
v0 = initial vertical velocity (0 m/s)
t = time taken
g = acceleration due to gravity (-9.81 m/s² in this case)
Rearranging the equation to solve for time:
t = sqrt(2h / g)
Substituting the given values:
t = sqrt(2 * 8.64 m / 9.81 m/s²)
Simplifying the equation:
t = sqrt(17.28 m / 9.81 m/s²)
t = sqrt(1.763 s²)
t ≈ 1.33 s
Therefore, it takes approximately 1.33 seconds for the compass to hit the ground.
To determine the time it takes for the compass to hit the ground, we need to consider the motion of the balloon and the motion of the compass.
Let's start by analyzing the motion of the balloon. We are given that the balloon is rising upward with a constant speed of 2.83 m/s. This means the vertical velocity of the balloon is +2.83 m/s.
Next, we need to consider the motion of the compass after it is dropped. Since the compass is no longer supported by the balloon, it will experience free-fall under the influence of gravity. The acceleration due to gravity is approximately -9.8 m/s^2 (negative because it acts downward).
Now, let's apply the kinematic equations to determine the time it takes for the compass to hit the ground.
We can use the following kinematic equation to find the time of flight:
y = y0 + v0*t + (1/2) * a * t^2
In this equation:
- y is the final position of the compass (0 m, as it hits the ground)
- y0 is the initial position of the compass (8.64 m, above the ground)
- v0 is the initial velocity of the compass (0 m/s, as it is dropped from rest)
- a is the acceleration due to gravity (-9.8 m/s^2)
- t is the time we want to find
Plugging in the values, we have:
0 = 8.64 + 0*t + (1/2)*(-9.8)*t^2
Simplifying the equation, we get:
4.9*t^2 = 8.64
Dividing both sides by 4.9, we have:
t^2 = 8.64 / 4.9
Taking the square root of both sides, we find:
t ≈ √(8.64 / 4.9)
t ≈ √1.763
t ≈ 1.33 seconds
Therefore, it takes approximately 1.33 seconds for the compass to hit the ground after it is dropped from the hot-air balloon.