THE PARABOLA

R. Shankar, The Institute of Mathematical Sciences, Chennai

Vikram, an elehant in the Mudumalai wild life sanctury and his friend Lakshmi were playing with a ball. Vikram was very skilled in handling the ball. He would catch it with his trunk, place it on his forehead and then roll it down his trunk. He could then adjust the shape of his trunk so that when the ball rolled off, it would go up and then come down exactly into Lakshmi's hands, wherever she was standing.
"Wow ! You really understand the parabola !" exclaimed Lakshmi. "What is that ?" came the puzzled reply. Lakshmi had just been studying falling bodies, Galileo's experiments and Newton's laws in school. She had found it fascinating and was more than willing to tell Vikram all about it. "It is the shape of the path that the ball takes, the shape of its trajectory". 
Vikram was still puzzled "But the path is different depending on where you are standing. When is it whatever you said it was ?" 
Lakshmi smiled. "A PARABOLA. And it is always a parabola no matter where I am standing." she explained, "Just like a circle is a circle whether it is big or small, a parabola is a parabola whether it is big or small. When I am standing nearby, it is a small parabola and when I am standing far away, it's a big one".
"You mean the shape of the path is the same and only the size changes ? I don't believe it. You will have to show me". So Lakshmi took the ball and threw it several times while Vikram watched from the side. But he was still not convinced. "It is not easy to make out the path", he complained, "The ball moves too fast and I cannot see the whole shape of its path at the same time".
So how could Lakshmi and Vikram picturise the path of the ball ? Lakshmi had a brilliant idea. She took a hose pipe which was watering a lawn nearby and slanted it up . The water went up and down forming a beautiful curve. "See, this is a parabola. When I tilt it up more it falls closer and it is a smaller parabola. When I tilt it less the water goes further and it is a bigger parabola". 
"Ah yes. I can see that the shapes of the two curves is somewhat similar. But how do I know that the ball moves in the same way as the water ? They are different things. Also I would like to see both the curves at the same time". 
There was only one hose pipe so this was a problem. So they thought and thought. "Somehow we need to mark the different positions of the ball," said Vikram, "But marking it in the air is impossible". "Oh, but we can mark it on graph paper and make a graph !" exclaimed Lakshmi, "Can you roll the ball exactly the same way many times ?". "No problem ! but why ?" came the answer. "Then each time we will stop the ball at different positions and mark it out on the graph paper. Come I'll show you what I mean".
First they collected the things that they needed. Graph paper, a pencil, a measuring tape, a step ladder, a board and some clay from near the river. Then they marked out the place where Vikram would stand with some stones. They put the step ladder some distance away, measured the distance from the stones and the step ladder and marked that distance on the x-axis of the graph paper. They covered the board with the clay Laksmi stood on the step ladder holing it firmly. Vikram went to his spot and rolled the ball. The ball hit the board and maked the position it hit on the clay. Vikram then walked up to the step ladder and with the measuring tape, measured the height from the ground to the mark made by the ball. They marked that point on the y-axis of the graph paper. Then, from that point they went along the x-axis till they reached the point representing the distance of the step ladder from the stones and marked it. So they had marked one position of the ball on the graph paper. They moved the step ladder a little distance away and repeated the procedure. They then had a second position of the ball marked on the graph paper. They did this many times and finally had about about twenty positions of the ball marked on the graph paper. They drew a curve joining these twenty points and had captured the path of the ball on the graph paper !
It was hard work but they persevered till they had several such curves. Then they stared at the curves and saw the similarity in the shapes. "Wow ! until now only my trunk understood the parabola. Now my brain has also a little understanding of it". 
You too can get a graph of the trajectory of a ball. Maybe you could use a foot ruler, some stands, plastacine, a small hose pipe and ball bearings. Do the experiments and send us your graphs. We will publish the best graphs that we get.