Chapter 2.2, Problem 65E

To determine

To identify: The kick that travels farther before hitting the ground and the kick that travels faster.

Answer to Problem 65E

Since $D_1=50>40=D$, second kick travels farther before hitting the ground.

Since $8>5$, first kick travels higher.

Explanation of Solution

Given Information: The height of the ball increases until it reaches a maximum height of 8 yards and 20 yards away from the player. The second kick is modelled by $y=x\left(0.4-0.008x\right)$

Calculation:

Maximum height that the ball reaches in first kick is 8 yards.

The distance between the player and the ball at its maximum height is 20 yards, so the total distance travelled by the ball during first kick can be calculated by multiplying the distance by 2.

Distance travelled by the ball is,

  $\begin{array}{c}D=20\times 2\\ =40\text{ yards}\end{array}$

• Here, D is the distance travelled by the ball in first kick.

Since the vertex represents the maximum height, the height of the second call is,

  $\begin{array}{c}y=x\left(0.4-0.008x\right)\\ =-0.008x^2+0.4x\mathrm{......}\left(1\right)\end{array}$

The x- coordinate of the vertex of the parabola of the equation $y=a{x}^2+bx+c$ is,

  $x=-\frac{b}{2a}$

On comparing the equation (1) with $y=a{x}^2+bx+c$ ,

  $\begin{array}{c}a=-0.008\\ b=0.4\\ c=0\end{array}$

The x- intercept is,

  $\begin{array}{c}x=-\frac{b}{2a}\\ =-\frac{0.4}{\left(-0.016\right)}\\ =\frac{0.4}{0.016}\\ =\frac{400}{16}\\ =\frac{100}{4}\\ =25\end{array}$

The height of the parabola is the y- coordinate of the vertex, which is,

  $\begin{array}{c}y=x\left(0.4-0.008x\right)\\ =25\left(0.4-0.008\times 25\right)\\ =25\left(0.4-0.2\right)\\ =25\times 0.2\\ =5\end{array}$

Therefore, the maximum height reached by the ball in the second kick is 5 yards.

The distance between the player and the ball at its maximum height is 25 yards, so the total distance travelled by the ball during first kick can be calculated by multiplying the distance by 2.

Distance travelled by the ball is,

  $\begin{array}{c}{D}_1=25\times 2\\ =50\text{ yards}\end{array}$

• Here, $D_1$ is the distance travelled by the ball in first kick.