Final answer:
The question asks how long it will take for a projectile launched upward to hit the ground. This physics problem is solved using a quadratic equation derived from the kinematic equation for projectile motion. The time is found by setting the height function to zero and solving for the positive time value.
Step-by-step explanation:
The problem involves using the kinematic equation for projectile motion to determine how long it will take for a projectile launched upwards to hit the ground. The provided height function is h(t) = -16t^2 + 32t + 128, which is a quadratic equation in the variable t (time). To find the time when the projectile hits the ground, we need to determine when the height h(t) equals zero.
To solve this, we set the height function to zero and solve for t:
-16t^2 + 32t + 128 = 0.
This is a quadratic equation, which we can solve using the quadratic formula: t = [-b ± sqrt(b^2 - 4ac)] / (2a). Substituting in our values we get: t = [-32 ± sqrt(32^2 - 4(-16)(128))] / 2(-16).
After performing the calculation, we will find two values for t, one positive and one negative. Since time cannot be negative in this context, we will select the positive value. This positive value will be the time it takes for the projectile to hit the ground after being launched. It is important to note that only physical solutions that make sense in the context of the problem should be considered, which in this case is the positive time value.