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All courses Math II · F-LE.6 27 of 73
Apply quadratic functions to physical situations such as projectile motion under gravity.

Identify the initial height from a projectile function

Problem
Identify the initial height in projectile function h(t)=-16t²+48t+5.
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Hint

Set \(t=0\) to find the starting height of the projectile.

In a projectile model, the initial height is \(h(0)\), which is the constant term when the function is expanded.

Solution walkthrough

01

Interpret initial height

\[\text{initial}~\text{time}~t=0\]

The initial height is the projectile's vertical position at the launch instant, so evaluate h at zero seconds.

02

Substitute the launch time

\[h(0)=-16(0)^2+48(0)+5\]

Every value comes from the supplied model, with 0 seconds substituted for t.

03

Simplify

\[h(0)=0+0+5=5\]

Both time-dependent terms vanish at launch, leaving the constant term.

04

State the result with units

\[\text{initial}~\text{height}=5~\text{feet}\]

The output h measures height, and the projectile coefficients use feet and seconds, so the linear unit is feet.

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Another way

  1. In h(t)=at²+bt+c, the vertical intercept and initial height are c.

!

Common mistake

Do not report 48 as the initial height. That coefficient represents initial velocity; the height at t=0 is the constant 5.