Polynomials & Functions
Real-World Equations: Object Thrown in the Air
The height of an object thrown upward is modeled by h(t) = −16t² + v₀t + h₀ (in feet), where v₀ is initial velocity and h₀ is initial height. Solve to find height at a given time, when it hits the ground, or when it reaches max height.
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Worked Examples
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Problem
A ball is thrown upward with velocity 48 ft/s from ground level. h(t) = −16t² + 48t. Find the height at t = 2 seconds.
Step-by-Step Solution
Substitute t = 2 into h(t) = −16t² + 48t.
h(2) = −16(2)² + 48(2)
h(2) = −16(4) + 96 = −64 + 96 = 32
Answer
Height = 32 feet at t = 2 seconds
Problem
Using h(t) = −16t² + 48t, when does the ball hit the ground?
Step-by-Step Solution
Set h(t) = 0: −16t² + 48t = 0
Factor: −16t(t − 3) = 0
t = 0 (launch) or t = 3 seconds (lands)
Answer
The ball hits the ground at t = 3 seconds
Problem
Using h(t) = −16t² + 48t, when does the ball reach maximum height and what is it?
Step-by-Step Solution
Max height occurs at vertex: t = −b/(2a) = −48/(2×−16) = −48/−32 = 1.5 seconds
h(1.5) = −16(1.5)² + 48(1.5) = −16(2.25) + 72 = −36 + 72 = 36 feet
Answer
Max height of 36 feet at t = 1.5 seconds
Problem
h(t) = −16t² + 64t + 6. What is the initial height of the object?
Step-by-Step Solution
Initial height is when t = 0.
h(0) = −16(0)² + 64(0) + 6 = 6 feet.
Answer
6 feet
Problem
h(t) = −16t² + 32t. At what two times is the object at height 0?
Step-by-Step Solution
Set h(t) = 0: −16t² + 32t = 0
Factor: −16t(t − 2) = 0
t = 0 (launch) or t = 2 seconds (lands)
Answer
t = 0 and t = 2 seconds
Your Turn — Practice Problems
Try all 5 problems on your own first. Write out your work — that's how it sticks!
💡 Tip: Don't peek at the answers until you've genuinely tried each one.
h(t) = −16t² + 64t. Find height at t = 1.
h(t) = −16t² + 64t. When does it hit the ground?
h(t) = −16t² + 64t. What is the maximum height?
h(t) = −16t² + 32t + 5. What is the initial height?
h(t) = −16t² + 80t. At what time is the ball at height 0 again?
h(t) = −16t² + 48t. Find height at t = 1.
h(t) = −16t² + 48t. What is the maximum height?
h(t) = −16t² + 96t. When does the object hit the ground?
h(t) = −16t² + 64t + 10. What is the initial height?
h(t) = −16t² + 32t. What is the maximum height?
h(t) = −16t² + 64t. Find height at t = 3.
h(t) = −16t² + 80t. What is the maximum height?
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