←
Boss lesson · The Golden Vertex
Exam-hall rules, last question
score 0

Exam-hall rules, last question

This is the boss. No calculator — today g=10 m/s2g = 10\ \mathrm{m/s^2}, stated on the paper, and every draw is chosen to fit in your head. One flare, one long problem, and every answer feeds the next: the moment of the peak (that is the VERTEX of h(t)=−5t2+v0t+h0h(t) = -5t^2 + v_0 t + h_0, where v0v_0 — v-nought — is the launch speed in m/s and h0h_0 the platform height in metres), then the height it buys, then the discriminant of the landing equation, then the landing itself (that is a ROOT). Write each line down: one early slip rides the parabola all the way into the water.

Two things to hold onto. Here a=−5a = -5, which is NEGATIVE — so the minus branch of the quadratic formula is the one that lands in the future, and the plus branch returns a phantom time from before the launch. And because the flare starts on a platform rather than at the water, the flight is NOT symmetric: it falls further than it climbed, so the landing time is more than twice the peak time. Everything in the chapter plays here. 5★ under par is the chapter crown.

h

h=−b2ah = -\frac{b}{2a}

  • hh= Vertex x-coordinate
  • aa= Coefficient of x²
  • bb= Coefficient of x
Vertex x-Coordinate of a Parabola solver →
k

k=c−b24ak = c - \frac{b^2}{4a}

  • kk= Vertex y-coordinate
  • aa= Coefficient of x²
  • bb= Coefficient of x
  • cc= Constant term
Vertex y-Coordinate of a Parabola solver →
Δ

Δ=b2−4ac\Delta = b^2 - 4ac

  • Δ\Delta= Discriminant
  • aa= Coefficient of x²
  • bb= Coefficient of x
  • cc= Constant term
Discriminant of a Quadratic solver →
x

x=−b+b2−4ac2ax = \frac{-b + \sqrt{b^2 - 4ac}}{2a}

  • xx= Root (plus branch)
  • aa= Coefficient of x²
  • bb= Coefficient of x
  • cc= Constant term
Quadratic Formula (Positive Root) solver →
no calculator in the boss room — the numbers are chosen to fit in your head