Math Quiz: Parabola Path

[Ether]

The center of a circle of radius=1 traverses the parabolic trajectory.. y = -0.0433*x².. from x=-5 to x=+5 in the XY plane, and as it does so it "sweeps out" a path with upper and lower boundaries.

Find the equations for the upper and lower boundary curves.

[Ether]

Hint: the answer is not

y = -0.0433*x2 + 1
and
y = -0.0433*x2 - 1

[Qbot2640]

I get:

Top curve: y = -0351x^2 + 1

Bottom curve: y = -.0433x^2 - 1

[Ether]

Originally Posted by Qbot2640

I get:

Bottom curve: y = -.0433x^2 - 1

Not. See my previous post.

Originally Posted by Qbot2640

I get:

Top curve: y = -0351x^2 + 1

I assume there's supposed to be a decimal point before the "0".

Nice try, but no cigar.

[Qbot2640]

Originally Posted by Ether

Not. See my previous post.



I assume there's supposed to be a decimal point before the "0".

Nice try, but no cigar.

My mistake...I had all three equations together, and grabbed your original as the "bottom". The top I got (as you corrected my missing decimal) is 0.0351x^2 + 1

The bottom I got was 0.0474x^2 -1

I calculated two normal lines and found points that were 1 unit away from points on your given original. Then calculated the quadratics from the two sets of three points (although the y intercepts being known, one really only needs two points).

[Ether]

Originally Posted by Qbot2640

My mistake...I had all three equations together, and grabbed your original as the "bottom". The top I got (as you corrected my missing decimal) is 0.0351x^2 + 1

The bottom I got was 0.0474x^2 -1

The top equation y = -0.0351x^2 + 1 is still not right (even with the required "-" sign added back in )

The bottom equation y = -0.0474x^2 - 1 (with the "-" sign added) is a very good approximation in the given range, but as you can see in the attached graph, it diverges outside that range.

I calculated two normal lines and found points that were 1 unit away from points on your given original.

You've got the right general idea.

Then calculated the quadratics...

Hint#2: the top and bottom boundary curves are not true quadratics.

[Ether]

Originally Posted by Ether

Hint#2: the top and bottom boundary curves are not true quadratics.

Graph of upper and lower boundary curves

[RyanCahoon]

Can I give the answers in parametric form?

Spoiler for parametric plot equations:

x_lower(t) = t - 0.0866 t / √(1 + (0.0866 t)²)
y_lower(t) = -0.0433 t² - 1 / √(1 + (0.0866 t)²)

x_upper(t) = t + 0.0866 t / √(1 + (0.0866 t)²)
y_upper(t) = -0.0433 t² + 1 / √(1 + (0.0866 t)²)

[Ether]

Originally Posted by RyanCahoon

Can I give the answers in parametric form?

You got it

Reps to you !

[Ether]

Is it possible to represent Ryan's parametric equations in explicit form y=f(x) ?

Or even implicit f(x,y)=0 ?

[Ether]

Originally Posted by Ether

Is it possible to represent Ryan's parametric equations in explicit form y=f(x) ?

Or even implicit f(x,y)=0 ?

Does anyone have access to Mathematica?

[Ether]

Any math/physics teachers want to give this a try?