You are designing the first ascent and drop for a roller coa

You are designing the first ascent and drop for a roller coaster. You want the slope of the ascent to be .8 and the slope of the drop to be -1.6. You will connect two straight stretches by part of the parabola: y=ax²+bx+c of width 100 units.

a) Certainly you don\'t want a sharp corner in your tracks at the points where the linear parts meet the parabola. This puts a condition on the tangent lines of the parabola-What\'s the condition?

b)Find a formula for the parabola.

Solution

a> there is a downward parabola which joins two straight lines .That is the parabola is in the middle
of the two lines . let the two lines be
y = L1(x)
and y=l2(x)
and let the joining points of the parabola and the straight lines be A and B .A and B are the transition points
Now for the track to be smooth there shall be no abrupt changes in the direction
=> We want the linear segments L1(x),and L2(x) to be tangent to the parabola at the transition points A and B. To simplify
the equation we\'ll take the origin at A.

b>
Parabola is y = ax^2 + bx + c
slope of the parabola using differentiation is y\' = 2ax + b

Now ,f(x) = ax^2 + bx + c. where f\'(0) = 0.8 & f\'(100) = -1.6

we have set the origin at A so c = 0
at point y=0 (A) we want the slope to be 0.8 so b = 0.8
at point y=100 (B) we want the slope to be -1.6 so
-1.6 = 2 * a * 100 + 0.8; a = -.012

Hence the formula for the parabola is :
y= -0.12x^2 + 0.8b