A ball is released from a hot air balloon moving downward with a velocity of
-10.0 meters/second and a height of 1,000 meters. How long did it take the ball to reach the surface of Earth? Given: g = -9.8 meters/second2.
h(t) = 1000 - 10t - 4.9t^2
Just solve for h=0
To find out how long it took for the ball to reach the surface of Earth, we can use the equation of motion for free fall:
h = ut + (1/2)gt^2
Where:
h = height (1,000 meters)
u = initial velocity (-10.0 meters/second)
g = acceleration due to gravity (-9.8 meters/second^2)
t = time (unknown in this case)
Rearranging the equation:
(1/2)gt^2 + ut - h = 0
This is a quadratic equation in terms of t. We can solve this equation using the quadratic formula:
t = (-b ± sqrt(b^2 - 4ac)) / 2a
For our equation:
a = (1/2)g = -4.9
b = u = -10.0
c = -h = -1,000
Substituting these values into the quadratic formula:
t = (-(-10.0) ± sqrt((-10.0)^2 - 4(-4.9)(-1,000))) / (2(-4.9))
Simplifying:
t = (10.0 ± sqrt(100 + 19,600)) / 9.8
t = (10.0 ± sqrt(19,700)) / 9.8
Now, we calculate the two possible values of t:
t1 = (10.0 + sqrt(19,700)) / 9.8
t2 = (10.0 - sqrt(19,700)) / 9.8
Using a calculator, we get:
t1 ≈ 10.23 seconds
t2 ≈ -104.24 seconds
Since time cannot be negative in this context, we discard the negative value. Therefore, the ball takes approximately 10.23 seconds to reach the surface of the Earth.