Chapter 6.3, Problem 46E

Solution

To find: the minimum distance between the two balls and when this minimum distance occurs.

The minimum distance is $6.60$ feet approximately and it occurs at $1.2$ seconds.

Given information:

Angle at which Linda tosses the ball: $44°$

Initial velocity of the ball when Linda tosses it: $45$ ft/sec

Height at which the ball is tossed: $5$ ft

Distance between the two players: $78$ ft

Angle at which Chris tosses the ball: $39°$

Initial velocity of the ball when Linda tosses it: $41$ ft/sec

Formula used:

  $x=\left(v_0\cos \theta °\right)t$ and $y=-16t^2+\left(v_0\sin \theta °\right)t+h$ .

Calculation:

Substitute $45$ for $v_0$ , $44°$ for $\theta$ in $x=\left(v_0\cos \theta °\right)t$ .

  $\begin{array}{l}x=\left(45\cos 44°\right)t\\ =45\left(0.719\right)t\\ =32.36t\end{array}$

Substitute $45$ for $v_0$ , $44°$ for $\theta$ in $y=-16t^2+\left(v_0\sin \theta °\right)t+h$ .

  $\begin{array}{l}y=-16t^2+\left(45\sin 44°\right)t+5\\ =-16t^2+45\left(0.695\right)t+5\\ =-16t^2+31.28t+5\end{array}$

Substitute $41$ for $v_0$ , $39°$ for $\theta$ in $x=\left(v_0\cos \theta °\right)t$ .

  $\begin{array}{l}x=78-\left(41\cos 39°\right)t\\ =78-41\left(0.78\right)t\\ =78-31.98t\end{array}$

Substitute $41$ for $v_0$ , $39°$ for $\theta$ in $y=-16t^2+\left(v_0\sin \theta °\right)t+h$ .

  $\begin{array}{l}y=-16t^2+\left(41\sin 39°\right)t+5\\ =-16t^2+41\left(0.63\right)t+5\\ =-16t^2+25.83t+5\end{array}$

The time at which the two balls are at minimum distance can be found graphically. It is approximately $1.2$ seconds.

The graph below shows the minimum distance between the two balls.

  

Linda’s ball: The $x$ and $y$ co-ordinates are $\left(38.83,19.50\right)$ .

Chris’s ball: The $x$ and $y$ co-ordinates are $\left(39.62,12.96\right)$ .

Substitute the values in the distance formula.

  $\begin{array}{l}\sqrt{{\left(38.83-39.62\right)}^2+{\left(19.50-12.96\right)}^2}\\ =\sqrt{0.6241+42.7716}\\ =\sqrt{43.3957}\\ \approx 6.60\end{array}$

The minimum distance is $6.08$ feet approximately and it occurs at $1.2$ seconds.