Hello to everyone that's reading this.

For this linear least-squares regression problem (typed below and also), I correctly find the value of g (which is what the problem statement wants to have found), but I was curious about the value of a 0 a0 (and that's what this entire thread is about).

__Problem statement__ (Alternatively, one can view this PDF: http://docdro.id/GmeGXNr):

"To measure g (the acceleration due to gravity) the following experiment is carried out. A ball is dropped from the top of a 30-m-tall building. As the object is falling down, its speed v is measured at various heights by sensors that are attached to the building. The data measured in the experiment is given in the table.

____________

x (m) | v (m/s) |

----------------------

0 | 0 |

----------------------

5 | 9.85 |

----------------------

10 | 14.32 |

----------------------

15 | 17.63 |

----------------------

20 | 19.34 |

----------------------

25 | 22.41 |

----------------------

In terms of the coordinates shown in the figure (positive down), the speed of the ball v as a function of the distance x is given by v^2 = 2gx. Using linear regression, determine the experimental value of g."

__The solution in the PDF__ (Alternatively, one can view this PDF: http://docdro.id/GmeGXNr):

"The equation v^2 = 2gx can be transformed into linear form by setting Y = v^2. The resulting equation Y = 2gx, is linear in Y and x with m = 2g and **b = 0**. Therefore, once m is determined, g can be calculated using g = m/s. The calculations are done by executing the following MATLAB program (script file):

Code:

clear all; clc;
x=[0 5 10 15 20 25];
y=[0 9.85 14.32 17.63 19.34 22.41];
Y=y.^2; X=x;
% Equation 5-13
SX=sum(X); SY=sum(Y);
SXY=sum(X.*Y);
SXX=sum(X.*X);
% Equation 5-14
n=length(X);
a1=(n*SXY-SX*SY)/(n*SXX-SX^2)
**a0=(SXX*SY-SXY*SX)/(n*SXX-SX^2)**
m=a1
**b=a0**
g=m/2

When the program is executed, the following values are displayed in the Command Window:

a1 = 19.7019

**a0 = 1.9170**

m = 19.7019

**b = 1.9170**

g = 9.8510

Thus, the measured value of g is 9.8510 m/s^2."

__Basically, what's I'd like to know is:__

Should the value of a_0 be 0 or 1.9170380952380952381? What "wins"? The a_0 = (Sxx Sy − Sxy Sx) / (n Sxx − (Sx )^2) formula or the zero term in v^2 = 2gx + 0? Also, if the value of a_0 should be 0, doesn't the formula for a_1 assume that the formula for a_0 will be used, such that if it is not, a_1 would be inaccurate or less accurate?

Any input would be GREATLY appreciated!