A hot-air balloon is rising upward with a constant speed of 2.35 m/s. When the balloon is 2.50 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?
To determine the time it takes for the compass to hit the ground, we need to use the equations of motion.
Let's break down the problem:
1. The initial height of the compass is 2.50 m above the ground.
2. The hot-air balloon is rising upward at a constant speed of 2.35 m/s.
3. We need to find the time it takes for the compass to reach the ground.
First, we can find the time it takes for the balloon to rise by dividing the initial height of the compass by the upward velocity of the balloon:
time = height / velocity
time = 2.50 m / 2.35 m/s
time ≈ 1.06 seconds
So, it takes approximately 1.06 seconds for the compass to rise to the height it was dropped from.
Next, we need to find the time it takes for the compass to fall back to the ground. Since the only force acting on the compass is gravity, and neglecting air resistance, we can use the equation for free fall:
height = (1/2) * gravity * time^2
Rearranging the equation to solve for time:
time = √(2 * height / gravity)
Now, we can substitute the values of height and the acceleration due to gravity:
time = √(2 * 2.50 m / 9.8 m/s^2)
time ≈ 0.71 seconds
Therefore, it takes approximately 0.71 seconds for the compass to fall back to the ground.
To find the total time elapsed, we add the time it takes for the balloon to rise (1.06 seconds) and the time it takes for the compass to fall back to the ground (0.71 seconds):
total time = time to rise + time to fall
total time ≈ 1.06 s + 0.71 s
total time ≈ 1.77 seconds
Hence, it takes approximately 1.77 seconds for the compass to hit the ground.