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All courses Math II · F-BF.1.a 14 of 73
Build quadratic and exponential functions from context using explicit formulas, recursive processes, or calculation steps.

Write a projectile height function from initial height and initial velocity

Problem
Write a quadratic height function h(t) for this projectile: A ball is thrown upward from 5 feet with initial velocity 48 ft/s.
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Hint

Use the projectile form \(h(t)=at^2+vt+h_0\), then match the gravity term, initial velocity, and starting height from the question.

In feet, the gravity term is \(-16t^2\), the linear term is the initial velocity, and the constant term is the starting height.

Solution walkthrough

01

Start with the projectile model

\[h(t)~=~-16t^2+\text{vt}+h_0\]

For vertical motion in feet, the quadratic coefficient is \(-16\) because of gravity.

02

Use the initial velocity

\[v~=~48\]

The question says the ball is thrown upward at \(48\) ft/s, so the linear term is \(48t\).

03

Use the starting height

\[h_0~=~5\]

The ball starts \(5\) feet above the ground, so the constant term is \(5\).

04

Write the full function

\[h(t)~=~-16t^2+48t+5\]

This combines the gravity term, the initial upward velocity, and the initial height into one quadratic model.

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Another way

  1. Check the finished model at t=0: h(0)=-16(0)²+48(0)+5=5, so the starting height matches. The linear coefficient 48 also matches the initial velocity from the question.

!

Common mistake

Using \(+16t^2\) instead of \(-16t^2\), even though gravity makes the parabola open downward.