Question 31653
Hello!
The discriminant of a quadratic equation of the form {{{A*x^2 + B*x + C}}} is

{{{B^2 -4*A*C}}}

In order for the equation to have two roots (ie, two values of x for which the result of the function is zero), this discriminant must be positive. In the quadratic equation you provide, we have:
A = -16
B = v0
C = 0

Therefore, the discriminant is:

{{{(v[0])^2 -4*(-16)*0 = v[0]^2}}}

Since v0 is greater than zero, then the disciminant is positive; so we know that the equation {{{-16t^2 + v[0]t = 0}}} has two solutions. So we know that there are two values for t that make the altitude equal to 0.

When the quadratic coefficient in a quadratic equation is negative (in this case, it's -16), then the maximum of the equation can be found at its vertex, whose formula is {{{-B/(2A)}}}. So we know that at time {{{t = -v[0]/(2*(-16))=v[0]/32}}}, the maximum altitude is attained.

In order to find the maximum altitude (what we've found is the TIME at which the maximum altitude is attained) we simply plug the t we found into the quadratic equation. So the maximum altitude will be:

{{{-16*(v[0]/32)^2 + (v[0]^2)/32}}}