Answered

A quarterback throws a football to a teammate. The football is 6.5 ft above the ground when it leaves the quarterback's hand. His teammate catches it 3.5 s later, at a height above the ground of 5 ft.



Projectile motion formula:

$\begin{array}{l}

h(t)=-16 t^2+v t+h_0 \\

h_0=6.5 \\

v=? \\

h=5 \text{ when } t=3.5

\end{array}$



Determine the value of $v$, rounded to the nearest whole number.

$v \approx$

Verified Answer :

  • Use the formula \(h(t) = -16t^2 + vt + h_0\) with known values to set up an equation.
  • Simplify the equation to isolate the term containing the initial vertical velocity \(v\).
  • Solve for \(v\) and round the result to the nearest whole number.
  • The initial vertical velocity is \(\boxed{56}\).

Explanation

  1. Analysis of the Problem

The problem describes the motion of a football using the standard projectile motion formula: \(h(t) = -16t^2 + vt + h_0\). We are given the following information:

  • The initial height of the football is \(h_0 = 6.5 \text{ ft}\).
  • The football is caught at a height \(h = 5 \text{ ft}\) at time \(t = 3.5 \text{ s}\).
  • The constant \(v\) represents the initial vertical velocity that we need to find.

Substituting the known values into the equation, we have:
\[5 = -16(3.5)^2 + v(3.5) + 6.5\]

  1. Isolating the Variable \(v\)

We now isolate \(v\) in the equation. First, calculate the squared term:
\[3.5^2 = 12.25\]
Substitute this back into the equation:
\[5 = -16(12.25) + 3.5v + 6.5\]
Perform the multiplication:
\[-16 \times 12.25 = -196\]
Now the equation is:
\[5 = -196 + 3.5v + 6.5\]
Simplify the constant terms on the right-hand side:
\[5 = -189.5 + 3.5v\]

  1. Solving and Finalizing the Calculation

To solve for \(v\), add \(189.5\) to both sides of the equation:
\[5 + 189.5 = 3.5v\]
\[194.5 = 3.5v\]
Now, divide by \(3.5\):
\[v = \frac{194.5}{3.5}\]
The calculation yields:
\[v \approx 55.5714...\]
Rounding this value to the nearest whole number, we get \(v = 56\). Thus, the initial vertical velocity is approximately \(56 \text{ ft/s}\).

Examples

This projectile motion model is fundamental in sports science and physics. For example, a quarterback or baseball player uses these calculations—consciously or unconsciously—to adjust their aim and the speed of their throw to ensure the ball lands in a specific target area at a certain height. Understanding the relationship between initial velocity, gravity, and time allows for more accurate athletic performance and is a cornerstone in fields ranging from sports coaching to aerospace engineering.

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