Parabola investigation

Question

Edit 4: I added the below picture for clarity

I'm trying to figure out how to find the angle between the red line and the blue line, but I have no idea how to start. (I have a feeling that this somehow involves physics? I have not taken physics past the grade 10 level so I don't really have a clue as to how to even start D:)

Someone willing to help?

Edit 3: After tinkering around and looking around, I'm now stuck on how to find the angle between the parabola and the x-axis.

Edit 2: I am doing HL math in the IB curriculum and we are expected to use similar to 1st year uni math for this exploration.

Edit 1: I was also wondering if there was such an equation that can be produced to describe the arc of the water from point x1 to x2, given that the y-value (height) of the start/end points are the same? or different?

I am not sure if I'm asking this right, but I was wondering if anyone had ideas as to what else I can further research about parabolas for a math investigation I am doing for class?

Right now, I am thinking of connecting parabolas to water fountains, and possibly trying to find the 'optimal' angle of a parabolic arc to produce the 'nicest' water fountain shape. Also, I was thinking of varying the angle of the parabola and investigating how that affects the shape. I just don't know if this is too elementary, as I am in a higher level math class and my teacher wants me to be more mathematically sophisticated.

Also, regarding my angle idea, I'm still trying to find out how to actually go about that :P

Does anyone have any suggestions as to what else can be investigated?

Also, I have looked into the angry birds game ideas and it is too common, so I can't do that.

thanks! :)

Answer 1

Parabolas:

• Projectile Motion: Projectile motion is a form of motion in which an object or particle (called a projectile) is thrown near the earth's surface, and it moves along a curved path under the action of gravity only.
• Trajectory of object under action of inverse square law forces, also covers above point. Forces such as Electric, Magnetic gravitations
• The curve of the chains of a suspension bridge is always an intermediate curve between a parabola and a catenary, but in practice the curve is generally nearer to a parabola, and in calculations the second degree parabola is used.
• Parabolic Reflectors, usually we talk about circular or plane mirrors.This can be a good topic to research.
  • Parabolic Lenses. We talk of convex and concave lenses which in theory are perfectly circular. Maybe what'll happen if we pass light through a parabolic lense?

Answer 2

The parabola has $x$-intercepts $x = 0, 4$, so it must have equation $$y = a x (x - 4)$$ for some $a$. On the other hand, it passes through the point $(2, 4)$, and so we have $$(4) = a (2)[(2) - 4].$$ This forces $a = -1$, and hence the equation is $$y = -x^2 + 4x.$$

Now, the indicated line is the tangent line to the parabola at $x = 0$, so its slope is $$\left.\frac{dy}{dx}\right\vert_{x = 0} = (-2 x + 4)\vert_{x = 0} = 4.$$

Drawing a suitable right triangle and using the definition of the tangent function then shows that the indicated angle is $$\color{#bf0000}{\boxed{\arctan 4}} \approx 1.326 \textrm{ rad} \approx 75.96^{\circ} .$$