A hot-air balloon is rising upward with a constant speed of 2.50 m/s. When the balloon is 3.00 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?
Solve this quadratic equation for t:
y = 2.50t -4.9t^2 +3.00 = 0
Take the positive one of the two roots.
To find the time it takes for the compass to hit the ground, we can use the equation of motion for free fall:
h = (1/2) * g * t^2
where h is the height, g is the acceleration due to gravity (approximately 9.8 m/s^2), and t is the time.
In this case, the initial height of the compass is 3.00 m above the ground, and it will fall down due to gravity.
First, we need to find the time it takes for the balloon to rise 3.00 m. Since the balloon is rising upward with a constant speed of 2.50 m/s, we can use the equation:
h = v * t
where h is the height, v is the velocity, and t is the time.
Rearranging the equation to solve for time, we have:
t = h / v
Substituting the given values, we get:
t = 3.00 m / 2.50 m/s
t = 1.20 s
So, it takes 1.20 seconds for the balloon to rise 3.00 m.
Now, we can use this time to calculate the time it takes for the compass to fall from the initial height of 3.00 m to the ground.
Using the equation of motion for free fall:
h = (1/2) * g * t^2
Substituting the values:
0 = (1/2) * 9.8 m/s^2 * t^2
Simplifying the equation:
4.9 m/s^2 * t^2 = 0
Since we are finding the time it takes for the compass to hit the ground, we only consider the positive time, so t = 0 is not a solution.
Simplifying further, we get:
t^2 = 0
t = ā0
t = 0 s
Therefore, it takes 0 seconds for the compass to hit the ground.
Since the time it takes for the compass to hit the ground is 0 seconds, it means that the compass will hit the ground immediately after it is dropped from the balloon.
To determine the time it takes for the compass to hit the ground, we can use the equation for free-fall motion:
s = ut + (1/2)gt^2
where:
s = distance fallen (3.00 m)
u = initial velocity (0 m/s)
g = acceleration due to gravity (-9.8 m/s^2, taking downward as positive)
t = time
Since the hot-air balloon is rising upward, the initial velocity of the compass is 2.50 m/s downward. So, the equation becomes:
3.00 = (2.50)t + (1/2)(-9.8)t^2
Simplifying the equation:
1/2(-9.8)t^2 + 2.50t - 3.00 = 0
We can now solve this quadratic equation to find the time it takes for the compass to hit the ground.