Complete Vertical Motion Model
The complete vertical motion model h = -16t² + v₀t + h₀ gives the height in feet of an object launched upward, where v₀ is the initial velocity (ft/s) and h₀ is the initial height (ft) — a core application in enVision Algebra 1 Chapter 8 for Grade 11. The -16t² term models gravitational acceleration (half of 32 ft/s²). For an object launched from a 32-foot platform at 96 ft/s, finding when it reaches 128 feet requires solving 128 = -16t² + 96t + 32, which simplifies to t = 3 ± √3 seconds. This model connects quadratic equations to real physics of thrown and launched objects.
Key Concepts
The height in feet, $h$, of an object shot upwards into the air with initial velocity, $v 0$, from an initial height $h 0$, after $t$ seconds is given by the formula: $$h = 16t^2 + v 0t + h 0$$.
Common Questions
What does each term in h = -16t² + v₀t + h₀ represent?
-16t² accounts for gravitational acceleration (32 ft/s² downward, halved in the kinematic formula). v₀t accounts for the initial upward velocity. h₀ is the height at t = 0.
An object is launched from 32 ft with initial velocity 96 ft/s. When does it reach 128 ft?
Set 128 = -16t² + 96t + 32. Rearrange: 16t² - 96t + 96 = 0, divide by 16: t² - 6t + 6 = 0. Using the quadratic formula: t = (6 ± √12)/2 = 3 ± √3 ≈ 1.27 s (going up) or 4.73 s (coming down).
Why does the vertical motion formula use -16 instead of -32?
The gravitational acceleration near Earth's surface is approximately 32 ft/s². In the kinematic height formula h = h₀ + v₀t - ½gt², the ½ × 32 gives 16, so the coefficient becomes -16.
What does it mean when h = 0 in the vertical motion model?
When h = 0, the object is at ground level. Solving -16t² + v₀t + h₀ = 0 gives the time(s) when the object lands, which is the total time of flight.
How do you find the maximum height using this model?
The maximum occurs at the vertex. The time of maximum height is t = v₀/32 (since t = -b/(2a) = -v₀/(2·(-16)) = v₀/32). Substitute back to find the maximum h value.