Question

part c - extended answers (10 marks each)

1. a ball has to be bounced into a hoop located at a height of 2 m.

then it has to bounce a second time and hit the target on the wall located 7 m from the thrower
(at x = 0).
a motion detector collects data on the movement of the ball.
the first bounce follows the path of a parabola. data collected tells you that the ball reached a
height of 1.5 m after a distance of 1.75 m and dropped back to a height of
1.5 m after a distance of 3.25 m.
a computer software calculated the vertical scale change “a” of this parabola to be -8/9.
the second bounce also follows the path of a parabola. the ball reached a maximum height of
1.6 m after it had covered a distance of 6 m.
determine the height of the target located on the wall at a distance of 7 m from the thrower.

Explanation:

Step1: First parabola vertex

First, find the vertex of the first parabola. The x - coordinate of the vertex of a parabola \(y = a(x - h)^2 + k\) for a parabola passing through \((x_1,y_1)\) and \((x_2,y_2)\) with \(y_1=y_2\) is \(h=\frac{x_1 + x_2}{2}\). Here \(x_1 = 1.75\), \(x_2=3.25\), so \(h=\frac{1.75 + 3.25}{2}=2.5\).

Step2: First parabola equation

The equation of the first parabola is \(y=-\frac{8}{9}(x - 2.5)^2 + k\). Substitute \(x = 1.75\), \(y = 1.5\) into the equation:
\(1.5=-\frac{8}{9}(1.75 - 2.5)^2 + k\)
\(1.5=-\frac{8}{9}(- 0.75)^2 + k\)
\(1.5=-\frac{8}{9}\times0.5625 + k\)
\(1.5=-0.5 + k\), so \(k = 2\). The first - bounce parabola is \(y=-\frac{8}{9}(x - 2.5)^2+2\). When \(y = 0\) (second - bounce starting point), \(0=-\frac{8}{9}(x - 2.5)^2+2\)
\(\frac{8}{9}(x - 2.5)^2=2\)
\((x - 2.5)^2=\frac{9}{4}\)
\(x-2.5=\pm\frac{3}{2}\). We take \(x=4\) (since \(x>2.5\) for the right - hand side root) as the starting point of the second - bounce parabola.

Step3: Second parabola equation

The second - bounce parabola has vertex \((h,k)=(6,1.6)\) and passes through \((x_0,y_0)=(4,0)\). The equation is \(y=a(x - 6)^2+1.6\). Substitute \(x = 4\), \(y = 0\)
\(0=a(4 - 6)^2+1.6\)
\(0 = 4a+1.6\)
\(a=-0.4\). So the second - bounce parabola equation is \(y=-0.4(x - 6)^2+1.6\)

Step4: Calculate height at \(x = 7\)

Substitute \(x = 7\) into \(y=-0.4(x - 6)^2+1.6\)
\(y=-0.4(7 - 6)^2+1.6\)
\(y=-0.4\times1+1.6\)
\(y = 1.2\)

Answer:

\(1.2\) m