A hot-air balloon is rising upward with a constant speed of 3.06 m/s. When the balloon is 9.31 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 equation for t:
height = 0 = 9.31 + 3.06 t - (g/2) t^2
= - 4.9 t^2 + 3.06 t + 9.31
There will be two answers since this is a quadratic equation. Take the positive one.
t = (-1/9.8)[-3.06 - sqrt[(3.06)^2 + 182.5)]
= 1.726 s
To solve this problem, we can use the equation of motion for an object in free fall. The equation is:
h = (1/2) * g * t^2
Where:
h = height of the object
g = acceleration due to gravity (approximately 9.8 m/s^2)
t = time taken for the object to fall
In this case, the hot-air balloon is rising upward with a constant speed, so the compass will be subject to the acceleration due to gravity downwards.
First, we need to calculate the time it takes for the compass to reach its highest point. We can use the formula:
t = h / v
Where:
t = time
h = height (9.31 m)
v = velocity (3.06 m/s)
Plugging in the given values, we have:
t = 9.31 m / (3.06 m/s) = 3.04 seconds
So, it takes approximately 3.04 seconds for the compass to reach its highest point.
Next, we need to find the time it takes for the compass to fall back to the ground. Since the initial velocity is 0 m/s (at the highest point), we can use the equation:
h = (1/2) * g * t^2
Plugging in the values, we have:
9.31 m = (1/2) * 9.8 m/s^2 * t^2
Simplifying, we get:
t^2 = (2 * 9.31 m) / 9.8 m/s^2
t^2 = 1.9 seconds^2
Taking the square root of both sides, we find:
t = √(1.9 seconds^2) = 1.38 seconds
So, it takes approximately 1.38 seconds for the compass to fall back to the ground.
Finally, to find the total time that elapses before the compass hits the ground, we add the time it takes to reach the highest point (3.04 seconds) and the time it takes to fall back to the ground (1.38 seconds):
Total time = 3.04 s + 1.38 s = 4.42 seconds
Therefore, it will take approximately 4.42 seconds for the compass to hit the ground.
To determine the time it takes for the compass to hit the ground, we can use the equation of motion for vertically upward motion:
h = ut + (1/2)gt^2
where:
h = height of the balloon above the ground = 9.31 m
u = initial velocity = 0 m/s (since the compass was dropped)
g = acceleration due to gravity = 9.8 m/s^2
t = time
Since the balloon is rising with a constant speed, the initial velocity of the compass is equal to the upward velocity of the balloon, which is 3.06 m/s.
Substituting the values into the equation, we have:
9.31 = (3.06)t + (1/2)(9.8)(t^2)
Rearranging the equation to form a quadratic equation:
4.9t^2 + 3.06t - 9.31 = 0
We can solve this equation to find the value of t.
Using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
Where:
a = 4.9
b = 3.06
c = -9.31
Substituting the values into the formula:
t = (-3.06 ± √(3.06^2 - 4 * 4.9 * -9.31)) / (2 * 4.9)
Simplifying the equation:
t = (-3.06 ± √(9.3636 + 182.2972)) / 9.8
t = (-3.06 ± √191.6608) / 9.8
Taking the positive value (since time cannot be negative):
t = (-3.06 + √191.6608) / 9.8
t ≈ 1.038 seconds
Therefore, it takes approximately 1.038 seconds for the compass to hit the ground.