Quadratic Function for braking distanceI have this question about finding a quadratic equation:Car Velocity v (km/hr) : a series of non zero positive valuesCar Brake distance d (meter) : also a series of non zero positive values.The problem asks why a quadratic equation is suitable to find the relationship between v and d?The problem also asks if d = a v 2 + b v + c where a, b and c are constants, find the function.Here are v values (km/hr) : 20, 30, 40, 50, 60, 70, 80, 90, 100 Here are d values (meters) : 15, 20, 25, 40, 50, 65, 80, 100, 115My observation so far: I have seen a range of x values including zero value and corresponding y values being converted to quadratic equation using 1st difference and 2nd difference of y values. That solution may not work all the time, especially when I don't have zero value for x.Secondly, I think the problem lacks key information such as deceleration, response time.

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Quadratic Function for braking distanceI have this question about finding a quadratic equation:Car Velocity v (km/hr) : a series of non zero positive valuesCar Brake distance d (meter) : also a series of non zero positive values.The problem asks why a quadratic equation is suitable to find the relationship between v and d?The problem also asks if   d  =  a      v    2    +  b  v  +  c where a, b and c are constants, find the function.Here are v values (km/hr) : 20, 30, 40, 50, 60, 70, 80, 90, 100 Here are d values (meters) : 15, 20, 25, 40, 50, 65, 80, 100, 115My observation so far: I have seen a range of x values including zero value and corresponding y values being converted to quadratic equation using 1st difference and 2nd difference of y values. That solution may not work all the time, especially when I don&#039;t have zero value for x.Secondly, I think the problem lacks key information such as deceleration, response time.

snowman8842 · Accepted Answer

Step 1This is just a quick answer, but you have set of values   (      v    i    ,      d    i    ) and want to fit it to an equation of the form   d  (  v  )  =  a      v    2    +  b  v  +  c. Each term will have an associated error       r    i    =  d  (      v    i    )  −      d    i  . You want to minimise this for all terms, in other words:  min  ∑  (      r    i        )    2  To do this we can set up a system, say:  S  =  ∑      r    i    2    =  ∑  (  a      v    i    2    +  b      v    i    +  c  −      d    i        )    2  and to find the minimas we want all of the partial derivatives equal to zero:  0  =            ∂      S              ∂      a        =  2  ∑      v    i    2    (  a      v    i    2    +  b      v    i    +  c  −      d    i    )  0  =            ∂      S              ∂      b        =  .  .  .  0  =            ∂      S              ∂      c        =  .  .  .Step 2Hopefully you get the picture. Now rearrange all of said equations, I will show the top one:  a  ∑      v    i    4    +  b  ∑      v    i    3    +  c  ∑      v    i    2    =  ∑      d    i        v    i    2  You now have a value mutliplied by a,b,c which you can work out and using all 3 possible equations get something of the form:      [                            α                          β                          γ                                      δ                          ϵ                          ζ                                      η                          θ                          ι                      ]        (                            a                                      b                                      c                      )    =      (                            κ                                      λ                                      μ                      )  you will know all of the greek letters&#039; values so now just find the inverse of the matrix (online calculators for this exist) and you now have values for a,b,c

Quadratic Function for braking distance. Car Velocity v (km/hr) : a series of non zero positive values. Car Brake distance d (meter) : also a series of non zero positive values. The problem asks why a quadratic equation is suitable to find the relationship between v and d?

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