Question

How is gravity a quadratic function?

Answer 1

Refer to explanation.

Explanation:

We can apply quadratic functions to objects that are in motion under gravity. For this explanation, we take a look at one of the equations of motion from physics that itself is a quadratic function and we set the acceleration of an object as being influenced by gravity.

Recall that a quadratic equation looks like the following:

`f(x)=ax^2+bx+c`

If we were to recall one of equations of motion from physics, such as the equation:

`x_f-x_0=v_0*t+1/2a*t^2`

where, `x_f-x_0=`change in position of `x` `=Deltax`

and if we consider our acceleration, `a`, to be the gravitational acceleration `g`, then

`Deltax=v_0*t+1/2g*t^2`

rearranging this equation gives

`Deltax=1/2g*t^2+v_0*t`

and making the change in the position of `x` with respect to time `t` gives

`Deltax(t)=1/2g*t^2+v_0*t`

the `1/2g` part is like the `a` part in `f(x)=ax^2+bx+c`

the `v_0` part is like the `b` part in `f(x)=ax^2+bx+c`

`c` is just some constant (some number), so think of `c` in the equation `Deltax(t)=1/2a*t^2+v_0*t`
as being equal to `0`, `(c=0)`

the `t`'s are like the `x`'s in `f(x)=ax^2+bx+c`

So, if we look at them together:

`f(x)=(a)x^2+(b)x+c`

`Deltax(t)=(1/2a)t^2+(v_0)t`