A hot-air balloon is rising upward with a constant speed of 2.22 m/s. When the balloon is 7.74 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 find 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 in relation to the ground.
First, let's find the time it takes for the balloon to rise 7.74 m using the formula:
time = distance / speed.
The distance the balloon rises is 7.74 m, and the speed of the balloon is given as a constant 2.22 m/s.
So, time = 7.74 m / 2.22 m/s = 3.5 seconds.
Now, we need to determine the time it takes for the compass to fall from the balloon to the ground.
Since it is dropped, the compass experiences free fall, accelerated by gravity. The distance it falls is the same as the distance the balloon rises, which is 7.74 m.
We can use the equation for free fall to find the time:
distance = 1/2 * acceleration * time^2.
The acceleration due to gravity is approximately 9.8 m/s^2.
Therefore, 7.74 m = 1/2 * 9.8 m/s^2 * time^2.
Simplifying the equation, we get:
7.74 m = 4.9 m/s^2 * time^2.
Dividing both sides of the equation by 4.9 m/s^2:
time^2 = 7.74 m / 4.9 m/s^2,
time^2 = 1.58,
time ā ā1.58 ā 1.26 seconds.
So, it takes approximately 1.26 seconds for the compass to hit the ground after it is dropped from the balloon.