Final answer:
To calculate the time until the object hits the ground, a kinematic equation is utilized where the initial velocity, acceleration due to gravity, and initial height are known. After setting up the equation with the proper signs, the quadratic formula is applied to find the positive value of time, which represents the duration until impact.
Step-by-step explanation:
To determine how many seconds until the object hits the ground, we can use the kinematic equation for uniformly accelerated motion, which is:
s = ut + 0.5at^2
where:
- s is the displacement (height the object falls, in feet)
- u is the initial velocity (64 ft/s upwards, take as negative because the displacement will be downwards)
- a is the acceleration due to gravity (we use -32 ft/s^2, as gravity pulls downwards)
- t is the time in seconds
We also know the initial height h from which the object is launched, which is 80 feet. The equation now becomes:
-80 = -64t - 0.5(32)t^2
Solving this quadratic equation for t will give us the time it takes for the object to hit the ground. You can use the quadratic formula to find the roots of the equation:
t = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in our values:
- a = -16 (half of -32 ft/s^2)
- b = -64 (initial velocity, taken as negative)
- c = -80 (initial height, which is the displacement the object must cover to reach the ground)
You'll get two possible values for t, but you'll only consider the positive value as time cannot be negative.