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# How can you graph the path of a curling stone?

Updated: 10/24/2022

Newpond17

Lvl 1
12y ago

The path of a curling stone is a parabola. Any three non-collinear points determine a parabola. If you throw a ball into the air, the shape of its path will be a parabola, provided the wind doesn't blow it away and you don't throw it so hard that it goes into space. The Cartesian Coordinate System form of the equation of a parabola is...

y = ax2 + bx + c

You can call the point that the curling stone is released, (i,j), where "i" is the positive (right) or negative (left) distance from the midpoint of the "foul line" (in curling, you have to throw the stone from behind the foul line), and "j" is the negated distance behind the foul line where you throw the stone. Because the stone "curls", or curves, at some point, you could call the point where that happens (k,l). When the stone loses all of its momentum, and it stops, you could call the point where it stops (m,n). Plot the points on a graph, and connect them with a smooth curve. Three points is the minimum, but if you optionally use more, then it will be easier to draw the curve. Solve the system of equations

i = aj2 + bj + c

k = al2 + bl + c

m = an2 + bn + c

for a, b, and c, to give you the equation y = ax2 + bx + c for the parabola.

But that isn't much fun until you learn what a, b, and c represent.

A is the slope of the parabola (whether it is almost flat or whether it is very tall). If a is positive, the parabola looks like the letter capital letter U. If a is negative, the parabola looks like the upside-down capital letter U.

B is the diagonal lower-left or lower-right shift. If b is positive, then the parabola is shifted to the lower-left. If b is negative, then the parabola is shifted to the lower-right.

C is the vertical shift. If c is positive, then the parabola is shifted up. If c is negative, then the parabola is shifted down.

This is all advanced algebra stuff, and if you're not intelligent enough to solve that system of equations, then the equation for the parabola won't help you much. :)

Curling stones do not exhibit parabolic trajectories. They move in a plane which means that the dot product of the velocity vector and gravitational acceleration is zero. It is not Projectile motion!

It is also false that 3 non co-linear points establish a parabola. Non co-linearity simply means that the function is not a line. This being said, the function could be a hyperbola, semi circle, sine wave ect. Also "A" alone does not determine the slope of the parabola. the slope is given by the value of the derivative, which according to power rule is 2aX+b.

If I had to speculate as the function that best describes the trajectory of a curling stone, I would choose an exponential model since the stones tend to curl more as they approach the end of their path.

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12y ago

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