What is a parabola?

0:47 Welcome to Matters Not Boring by kidopoly.com. I'm Mira! And I'm Finn! Before we start, we want to say a huge hello to eight-year-old Frank, who is listening to us right now in Wheaton, Illinois. Hi Frank!

1:02 Hi Frank! It's so awesome to have you listening with us today. If you haven't already, tap follow wherever you're listening, so you never miss an episode.

1:12 And Frank, before we jump into today's idea, we have to mention our sister show, Science is Not Boring. Have you heard our episode called What is a Microwave?

1:23 Oh yeah! In that one, we actually look inside a microwave oven to see how invisible electromagnetic waves reflect off metal walls, pass right through the glass plate, and heat up your food by vibrating tiny water molecules.

1:37 It's such a fun show because we really love learning about all the amazing hidden science that is happening right in our very own kitchens. So we hope you all go listen and follow science is not boring too. But wait, what shape are we talking about today? Where are we going? Frank, I want you to imagine you are standing in a cold dark field at night, looking up at a massive professional fireworks display.

2:04 A huge rocket shoots up and explodes into a giant umbrella of glowing light.

2:10 Oh, I love those. When all the bright little sparks rain down everywhere.

2:15 Exactly, but here is the secret hiding in the sky. As every single one of those bright sparks falls back down to Earth, it traces out the exact same mathematical curve. Wait, really? Every single spark makes the same shape, but they are blowing all over the place. Every single one traces the same family of curves, and it is the exact same shape you make when you throw a ball to a friend.

2:41 Okay, throwing a ball. I do that all the time. But I guess I never really thought about the invisible line it draws in the air.

2:48 Well, let's picture it. Frank, imagine throwing a tennis ball to Finn. It leaves your hand and it rises up toward the sky. But it slows down. It kind of turns over at the very top.

3:02 And then it falls down in this beautiful, smooth, even arch, right into my hands.

3:07 Yes, that smooth arch has a real mathematical name. It is called a parabola. A parabola? It sounds like a magic spell. Parabola! It totally does. Ancient mathematicians knew about parabolas, but the path of a thrown ball was a mystery. We only realised they draw parabolas in the year 1638, thanks to an amazing Italian scientist named Galileo Galilei.

3:35 Galileo?

3:36 The guy who used telescopes to look at the moon and the stars.

3:39 That's him. In his last great book, he wasn't looking at the stars. He was puzzling over how things move right here on Earth. He was actually forced by the authorities to stay inside his house at the time, but his mind never stopped exploring the universe.

3:55 So he's stuck inside his house, thinking about what happens when you throw things.

3:59 Exactly. And you have to understand, before Galileo, people had some totally wild ideas about how throwing worked. They actually thought a heavy cannonball would shoot straight out in a perfectly flat line, magically run out of forward energy, stop completely in mid-air, and then drop straight down like a stone.

4:20 Wait, nobody actually thought that. That can't be right. Have they never thrown a rock? You can literally see it curving.

4:27 They did. But when things move fast, human eyes aren't great at tracing the exact path. Galileo was the first one to prove that anything thrown, a ball, a rock, or a drop of water, draws a parabola.

4:41 Okay, but how do they figure it out if it happens too fast to see?

4:45 Galileo figured out the true curve by splitting the throw into two separate movements that are happening at exactly the same time. Imagine the throw carrying the ball forwards.

4:56 Okay, so if I throw it, it glides forward. And if nothing was pulling on it, it would just keep flying forward at a steady speed forever.

5:05 Yes! But something is pulling on it. Gravity! At the exact same moment the ball is drifting steadily forwards. Gravity is pulling it down. And gravity doesn't just pull at a steady speed, it pulls things down faster and faster every second.

5:23 Oh wow! So it's trying to move forward steadily, but it's dropping faster and faster at the exact same time.

5:30 Exactly! And when you combine a steady forward glide with a faster and faster downward fall, they blend together to create that beautiful sweeping arch, the parabola.

5:41 But hold on, what if you're throwing a really light ball on a windy day? Does the wind mess up the perfect math?

5:48 That is such a smart question. A pure perfect parabola is what happens if we completely ignore air resistance. But in the real world, real air pushes back against the ball. So the air kind of squeezes the shape. Right. The air makes the true path just a tiny bit squashed. Like with those big fireworks at organized displays.

6:10 The glowing sparks slow down really quickly as they push through the air. So they draw parabola-like arcs instead of perfect ones. But the maths is always right there inside them.

6:22 That's so cool. So every single time I throw a tennis ball across the yard, gravity and forward speed are drawing a parabola together. It is cool.

6:32 But here is the part that is going to bend your brain a little. Because this idea connects sideways. The parabola works the other way round too.

6:41 What do you mean the other way around? Like catching things instead of throwing them?

6:46 Exactly like catching. Think of a satellite dish. You know those big curved metal dishes you see pointing up at the sky, attached to the sides of houses or at space stations?

6:58 Oh yeah. They look kind of like giant shallow bowls. I always wondered why they were shaped like that.

7:05 They are shaped like parabolas. If you take a parabola curve and spin it around in a circle, it makes that exact bowl shape. And that shape does something completely magical. Magical how? It's just a metal bowl. When invisible signals travel through space and hit the curved inside of a parabolic dish, the curve bounces every single parallel wave coming straight in to exactly one single spot.

7:34 No way! It bounces all of them to the exact same point. Even if they hit the edge of the dish.

7:41 Every single one. No matter where the straight signal hits the curve, it gets angled directly into the exact same dot in the air. That point is called the focus.

7:52 Oh! That's why satellite dishes have that little plastic receiver box hanging on a metal arm right in the middle. It sits perfectly at the focus, catching every single wave the dish bounces.

8:05 That is the secret! If the dish was just a normal circle bowl, the waves would bounce all over the place and make a messy blur of a signal. The parabola is the only shape that gathers parallel lines perfectly into one focus.

8:20 So this one mathematical idea, this parabola, is the shape of a ball flying across a park. And it's also the exact shape needed to catch invisible signals from outer space.

8:32 Yes! Once you see it, you start spotting parabolas everywhere. If you watch a jet of water splashing out of a drinking fountain, that smooth curve of water is a parabola.

8:44 And when you watch those giant rockets at a professional bonfire night display, the sparks trace out parabola-like arcs as they fall.

8:52 Exactly. All hiding in plain sight. And here is a mind-bending fun fact about Galileo, discovering all this back in 1638. Did he get in trouble for figuring out throwing? Well, his books were banned by the authorities in Italy at the time. He had to have the manuscript for his greatest book secretly smuggled all the way to a publisher in the Netherlands, just so the world could read about parabolas.

9:18 Wait, he actually smuggled a math book out of the country? That makes him sound like a secret agent.

9:25 He was just completely determined to share what he saw. He knew he had found a deep truth about how the world moves, and he wouldn't let it stay hidden. Alright, Finn, it's time for the quiz! Are you ready? I'm ready. Bring it on! First question. What is the real mathematical name for the smooth art shape we just talked about?

9:50 Next one. Which famous Italian scientist wrote the book about this shape? Question three. What year did he publish his great book about it? Here's number four. What does gravity make the downward speed do as it pulls on a thrown ball? Last question. What do we call the single point where a satellite dish focuses all its invisible signals?

10:29 Great job! Let's see how you did. The first answer is the curve is called a parabola. For question two, the scientist was Galileo Galilei. The answer to number three is the book was published in 1638. For question four, gravity makes it drop faster and faster every second. And the final answer is that single exact spot is called the focus.

10:58 I love that we started just throwing a ball to Frank in Illinois and ended up with smuggled spy books and space signals bouncing into dishes. That is the beauty of maths.

11:09 It connects absolutely everything. If you loved finding out about parabolas with us today, please leave the show a five-star review.

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11:25 And be sure to tap follow so you get every new idea. Oh, and we have sister shows too! History, science and philosophies, not boring. We would really love for you to give those a listen as well.

11:39 And if you want to shout out on the show, or just want to say hi, email us at hello at kidopoly.com.

11:46 Thanks for exploring the shape of the world with us today. See you next time on Maths is not boring. Bye everyone.