Chapter SH, Problem 9.7EP

To determine

To find the equation of the parabolic path jump.

Answer to Problem 9.7EP

  $y=-\frac{1}{24}{x}^2+6$

Explanation of Solution

Given:

A long jumper makes a leap of $24$ feet with maximum height of $6$ feet

Calculation:

Consider a long jumper makes a leap of $24$ feet with maximum height of $6$ feet. Thus, the path of the long jumper can be modelled by a parabolic path and consider that the focus of the parabola is at origin, the figure is given below:

  

Therefore,

  $BC=24,OA=6$

As the parabola is downwards therefore consider the equation of the path as:

  $y=-a{(x-h)}^2+k$

As $OA$ is the maximum height hence it’s the vertex and therefore:

  $\begin{array}{l}h=0\\ k=OA\\ k=6\end{array}$

Therefore the equation of the path is written as:

  $y=-a{x}^2+6$

As the path intersects the $x$ axis at $C(12,0)$ therefore:

  $\begin{array}{l}0=-a{(}^{12}+6\\ a=\frac{6}{144}\\ a=\frac{1}{24}\end{array}$

Therefore the equation of the parabolic path is:

  $y=-\frac{1}{24}{x}^2+6$