101 uses of a quadratic equation: Part II

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08 Nov 10

Jeffrey Cole
101 uses of Quadratic Equation
07 Nov 10

Winnie Beattie
101 applications of the quadratic equation
14 Jan 10

William Hummel
- If an object is moving in one direction without a force acting on it, then it continues to move in that direction with a constant velocity. We can call this velocity $v$ . Now, if the particle starts at the point $x=0$ and moves in this way for a time $t$ , then its resulting position is given by $x = vt.$ Usually the particle has a force acting on it, such as gravity for a rugby ball or friction in the brakes of a car. Fast-forwarding to Newton we know that the effect of a constant force is to produce a constant acceleration $a$ . If the starting velocity is $u$ , then the velocity $v$ after a time $t$ is given by $v = u + at$ . Galileo realised that you could go from this expression to working out the position of the particle. In particular, if the particle starts at the position $x=0$ then the position $s$ at the time $t$ is given by
  
  $s = ut + \frac{1}{2} at^2.$
  This is a quadratic equation linking $t$ to $s$ with many major implications for all of us. For example, suppose that we know the braking force applied to a car: then this formula allows us to work out either how far we travel in a time $t$ , or conversely, solving for $t$ , how long it takes to travel a given distance.
- The simple quadratic formula relating time to distance is also the basis of the science of ballistics, which looks at the way that objects move under gravity. In this case, an object falls in the $y$ direction with a constant acceleration $g$ . In contrast, it moves in the $x$ direction horizontally at a constant velocity (in the absence of air resistance). If it starts at the point $x = y = 0$ with velocity $u$ in the $x$ direction and velocity $v$ upwards, then Galileo was able to show that the position at time $t$ is given by
  
  $x = ut \quad \mbox{and} \quad y = vt - \frac{1}{2} g t^2.$
  Put another way, we have
  $y = \left( \frac{v}{u} \right) x - \left( \frac{g}{2u^2}\right) x^2,$
  yet another quadratic equation, this time relating $x$ to $y$ . What was remarkable was that the resulting shape of the trajectory was, of course, a parabola.
18 Oct 09

Sameer Shah
math-resources algebra2
20 Oct 08

deepti dixit
Quadratics equations in real life
10 Aug 08

Heather A
A little more history of quadratics.

umb611
- Quadratic equations not only described the orbits along which the planets moved round the Sun, but also gave a way to observe them more closely.
31 Jul 08

Tania Kennedy
maths quadratics
17 Apr 08
04 Feb 07

Stephanie Disbury
Maths_Resources Maths_Websites
20 Nov 06

Andrew Demack
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