How is gravity a quadratic function?

1 Answer
Jul 13, 2015

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#