Jacob Birch

Railroad and Bicycle Problems

Railroad Curve ProblemCurves on railroad track are in the shape of cubic parabolas. Such “parabola tracks” have the property that curvature starts as zero at a particular point and increases gradually, easing the locomotive into the curve slowly so that it is less likely to derail. The track is defined by the piecewise function

Railroad Curve Problem

Our Task: Find the coefficients a,b,c and d in the cubic branch of the function.This will give us a smooth transition from the curve to the straight track and make the train less likely to derail. Then find the value of k in the linear branch of the function that makes the piecewise function continuous.

Railroad Curve ProblemThis graph shows what we the two functions should look like if the coefficients and k are found that make the transitions smooth. However, some of the function will be discarded.

Railroad Curve Problem

Railroad Curve Problem

Given: The curve’s left branch of the graph contains the origin and has y’=0 at that point. At the transition point where x=1/2 y’=1 so that the curve goes the same direction as the straight section. At this point y’’=0 so that the curvature is zero.

Railroad Curve Problem

Since we know we will need the derivative and second derivative equations so we find them. This will put the equations into our “toolbox” so we can use them when needed.𝑦={ {x + k, if x>

Y’=3𝑎+2𝑏 +𝑥 𝑐Y”=6ax+2b

Railroad Curve Problem

We know the piecewise function must contain the origin so we plug in the coordinates (0,0). To find the coefficients we will feature the top half of the piecewise function.

0=d

This “d” value is essentially a vertical translation component. It places the cubic graph on the origin.

Railroad Curve Problem

Given that y’=0 @x=0 we can plug (0,0) into the derivative equation. y’= 2𝑏𝑥+𝑐 � plug in (0,0) � simplify 0=0+0+c � simplify0=c

We have found the coefficient c which will help make the piecewise function transition smoothly.

Railroad Curve Problem

Given that y’=1 @x=1/2 we can plug (1/2,1) into the derivative equation and simplify.

y11=1==b

Railroad Curve ProblemGiven that y’’=0 @x=1/2 we can plug (1/2,0) into the second derivative equation and simplify.

y’’=6ax+2b � plug in numbers00=3a+2b � since we solved for b earlier we can now substitute for b0=3a+2(1- � simplifyProblem continued on next slide.

Railroad Curve Problem

0= 0= −2 � � /(3/2) We have found the coefficient a which will help make the piecewise function transition smoothly.

Railroad Curve ProblemWe can now plug a into the equation b=1-𝑎=b= 1- � simplifyb=1-(-1) � rewriteb=1+1b=2

We have found the coefficient b which will help make the piecewise function transition smoothly.

Railroad Curve ProblemWe must find the value of k which makes the piecewise function continuous. To do this set the two pieces of the piecewise equal to each other. But first plug in the coefficients a , b ,c, and d into the top half of the piecewise.

� simplify and set = to x + k

� simplifyProblem continued on next slide.

Railroad Curve Problem

�We have found k which will help make the piecewise function transition smoothly.

Railroad Curve Problem

Final Graph

𝑦={𝑎𝑥3+𝑏𝑥2+𝑐 𝑥+𝑑 , 𝑖𝑓 0≤ 𝑥≤12

𝑦={𝑥+𝑘 , 𝑖𝑓 𝑥>12

Bicycle Frame Design ProblemWe will now take a gander at the Bicycle Frame Design Problem.

Bicycle Frame Design Problem

A bicycle frame’s front fork, holds a wheel. To make the bike track properly, the fork curves forward at the bottom where the wheel bolts on. Assuming that the fork is bent in the shape of the cubic parabola, y= . What should the constants a and b be so that the curve joins smoothly to the straight part of the fork at the point (10,20) with slope equal to 5?

Bicycle Frame Design Problem

Bicycle Frame Design ProblemFor the parabolic function to connect to a linear function with a smooth transition we will need to find a linear function to put with the parabolic function and then put them together as a piecewise function.

To find a linear equation use the basic point-slope form of an equation and the fact that the linear function must contain the point (10,20) and have a slope of 5.

Work done on next slide.

Bicycle Frame Design Problem

y - y₁ =m(x - x₁) � plug in point (10,20) and slope 5y-20=5(x-10) � simplifyy-20=5x-50 � +20y=5x-30Put the linear function and cubic parabolic function together as a piecewise equation.

y={ {5x-30, if

Bicycle Frame Design ProblemFor y “the piecewise function” to be smooth it must be both continuous and differentiable @ x=10

To be continuous @ x=10 � plug in 10 � simplify � -1000𝑎 �/10𝑏=2100𝑎

Bicycle Frame Design ProblemTo be differentiable @ x=10 We must find the derivatives of the top and bottom of the piecewise and set them equal to each other.

y={ {5x-30, if ≥10𝑥y’=if x<10 {5, if ≥10𝑥 � plug in 10 � simplify � -300𝑎

Bicycle Frame Design Problem

We know that b = 2 – 100a and we know that b = 5 – 300a so we can set these equal to each other eliminating the b’s and solve for a.

� +300𝑎 � -2 � /200

Bicycle Frame Design Problem

Since we have found a we can plug it into the b = 5 – 300a or b = 2 – 100a equation and find b. I choose to use the first of the two equations.

� plug in a � simplify � simplify

Bicycle Frame Design Problem give us a smooth transition

𝑦={𝑎𝑥3+𝑏𝑥 , 𝑖𝑓 𝑥<10

y = {5x - 30, if ≥10𝑥