Jan is standing on a roof of a building. She launches a water balloon upward at an initial speed of 44 feet per second at height of 20 feet. How long will it take before the balloon lands on the ground? Round to the nearest tenth of a second.
the height h of the balloon above the ground is given by
h(t) = 20 + 44t - 16t^2
set h=0 and solve for t
To find the time it takes before the water balloon lands on the ground, we can use the equation for the vertical motion of an object:
h = h0 + v0*t + (1/2)gt^2
Where:
h is the final height (0 for the ground)
h0 is the initial height (20 feet)
v0 is the initial velocity (44 feet per second)
g is the acceleration due to gravity (32.2 feet per second squared)
t is the time in seconds that the balloon is in the air.
Plugging in the values, we get:
0 = 20 + 44t - 16.1t^2
Simplifying the equation gives:
16.1t^2 - 44t - 20 = 0
We can solve this quadratic equation for t using the quadratic formula:
t = (-b ± sqrt(b^2 - 4ac))/(2a)
where a = 16.1, b = -44, and c = -20.
Plugging in the values, we get:
t = (-(-44) ± sqrt((-44)^2 - 4(16.1)(-20)))/(2(16.1))
Simplifying this equation gives two solutions: t ≈ 0.85 seconds and t ≈ 2.07 seconds.
Since we're interested in the time it takes before the balloon lands on the ground, we need to take the positive value:
t ≈ 0.85 seconds.
Therefore, it will take approximately 0.85 seconds for the water balloon to land on the ground.
To determine how long it will take for the water balloon to land on the ground, we can use the concept of projectile motion and the equations of motion.
First, we need to find the time it takes for the water balloon to reach its highest point. The initial velocity of the water balloon is 44 feet per second, and it is launched upward, so we assume the acceleration due to gravity is acting against it. The acceleration due to gravity is approximately 32.2 feet per second squared.
We can use the following equation of motion to find the time it takes for the balloon to reach its highest point:
v = u + at
Where:
v = final velocity (0 ft/s at the highest point)
u = initial velocity (44 ft/s)
a = acceleration (-32.2 ft/s^2)
t = time
0 = 44 - 32.2t
Solving for t:
32.2t = 44
t = 44 / 32.2
t ≈ 1.366 seconds
It will take approximately 1.366 seconds for the water balloon to reach its highest point.
Next, we need to find how long it takes for the balloon to fall from its highest point to the ground. The height from which the balloon is launched is 20 feet, so the total journey downward to the ground will be 20 feet.
We can use the equation:
s = ut + 0.5at^2
Where:
s = distance (20 ft)
u = initial velocity (0 ft/s at the highest point)
a = acceleration (32.2 ft/s^2)
t = time
20 = 0 + 0.5 * 32.2 * t^2
Simplifying the equation:
10.1t^2 = 20
t^2 = 20 / 10.1
t^2 ≈ 1.98
t ≈ √(1.98)
t ≈ 1.4 seconds
It will take approximately 1.4 seconds for the water balloon to fall from its highest point to the ground.
Therefore, to find the total time it takes for the balloon to land on the ground, we add the time it takes to reach the highest point (1.366 seconds) and the time it takes to fall back down (1.4 seconds):
Total time ≈ 1.366 + 1.4
Total time ≈ 2.766 seconds
Rounded to the nearest tenth of a second, it will take approximately 2.8 seconds for the water balloon to land on the ground.