A quadratic process: the height of a ball

Science and society, grade 12 (3 units) Lesson 1 of 14Learning

A quadratic process: the height of a ball

A ball kicked upwards

A player kicks a ball straight up, from the level of the grass.
The ball’s height, in metres, $\displaystyle t$ seconds after the kick:

h=-5t^{2}+20t

We substitute a few times in the formula and get a table:

$\displaystyle t$ (seconds)	$\displaystyle 0$	$\displaystyle 1$	$\displaystyle 2$	$\displaystyle 3$	$\displaystyle 4$
$\displaystyle h$ (metres)	$\displaystyle 0$	$\displaystyle 15$	$\displaystyle 20$	$\displaystyle 15$	$\displaystyle 0$

For example, after one second: $\displaystyle {h}={-5\cdot1^{2}+20\cdot1}={-5+20}={15}$ metres.

The graph of the height

On the graph board,

\displaystyle x

is the time

\displaystyle t

, and

\displaystyle y

the height

\displaystyle h

The points of the table, and the curve through them:

\displaystyle y=-5x^{2}+20x

\displaystyle (0,0)

\displaystyle (1,15)

\displaystyle (2,20)

\displaystyle (3,15)

\displaystyle (4,0)

The graph is a parabola that opens downwards:

at first the ball rises fast – 15 metres in the first second;
it slows down – only 5 metres in the second second;
after 2 seconds it is at its greatest height, 20 metres;
it falls, and after 4 seconds lands on the grass.

The graph shows the height against time, not the ball’s path: the ball goes up and down along the same straight line.

A quadratic model

A quadratic model is a formula in which the variable appears squared, and in no higher power:

\displaystyle y=ax^{2}+bx+c

\displaystyle a\neq0

$\displaystyle a$ decides which way the parabola opens: $\displaystyle a<0$ – downwards, and it has a highest point; $\displaystyle a>0$ – upwards, and it has a lowest point.
$\displaystyle c$ is the value of $\displaystyle y$ when $\displaystyle {x}={0}$ : for the ball, the height it started from.
For the kicked ball $\displaystyle {a}={-5}$ , $\displaystyle {b}={20}$ and $\displaystyle {c}={0}$ . The $\displaystyle -5$ comes from gravity: it slows down anything flying upwards, and pulls it back.

The symmetry of the table

The ball’s table has the same height at two times: 15 metres after one second – on the way up, and after 3 seconds – on the way down.

These times are equally far from $\displaystyle {t}={2}$ , the time of the greatest height.
So the upright line $\displaystyle {t}={2}$ is the parabola’s axis of symmetry: the half to its left is the mirror image of the half to its right.
The highest point, $\displaystyle (2,20)$ , is the parabola’s vertex.

More quadratic processes

The same shape comes back in many phenomena:

a fountain’s jet – the water’s height against the distance from the nozzle;
the area of a rectangle with a fixed perimeter, against the length of one side;
a car’s braking distance against its speed: twice the speed – four times the braking distance;
a bridge’s arch – the arch’s height against the distance from its end.

Exercise 1

A girl throws a ball upwards from a balcony 15 metres high.
The ball’s height above the pavement, in metres, $\displaystyle t$ seconds after the throw:
$\displaystyle {h}={-5t^{2}+10t+15}$

Work out the height after one second, after 2 seconds and after 3 seconds.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power