Quadratics

6.1 Projectiles

We have already seen in part 1 of this course that any sequence of numbers that has constant double differences is given by a quadratic formula.

For example, the sequence 2, 3, 6, 11, 18, 27, 38, … has constant double differences:

Q6_1_pic1

We see that its leading diagonal is given as a combination of the standard leading diagonals:

Q6_1_pic2

This suggests that the sequence is given by the quadratic formula \(n^{2}-2n+3\) (and one can check that this works).

 

PRACTICE 118: Find a formula for the sequence 13, 26, 49, 82, 125, 178, 241, …

Recall that all solutions to the practice problems appear in the COMPANION GUIDE to this QUADRATICS course. 

 

And as we saw in that early section, Galileo used this kind of idea to deduce that because acceleration due to gravity is constant, and acceleration is a kind of double difference, all objects propelled through the air follow the paths of quadratic graphs.

COMMENT: Air resistance and wind, sadly, ruin this ideal observation. But as a first start to understanding the motion of objects it seems helpful to initially ignore these effects and study this ideal motion to make some first predictions.

 

PRACTICE 119: Lizzy throws a ball into the air. Its height, in feet, at time \(t\) seconds is given by the quadratic formula:

\(H(t)=-32t^{2}+480t+6\).

a) Find \(H(0)\). What does this number mean?

b) At what time is the ball at its maximum height?

c) When does the ball hit the ground?

 

COMMENT: As we saw in part I, Galileo also wondered if the shapes of hanging chains were also the shapes of quadratic curves.

Q6_1_pic3

It turns out they are not! These curves are called catenary curves (from the Latin word for “chain”) and are given by a complicated formula. Their mathematics were not properly figured out until 1691, once the subject of calculus was invented.

 

PRACTICE 120: (OPTIONAL)  Conduct some internet research to learn about catenary curves, catenary arches, and their history.

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