Page 159 - Data Science Algorithms in a Week

Regression

Where v is the initial velocity of the object, τ is an angle at which the object was fired and g
is the gravitational force exerted by the planet on the object. Note that the angle τ and the
gravitational force g do not change. Therefore define a constant . Then the
distance on the explored planet can be explained in terms of the velocity by the equation:

Although d and v are not in the linear relationship, d and the square of v are. Therefore we
can still apply the linear regression to determine the relationship between d and v.
Analysis using R:

Input:

source_code/6/speed_distance.r
trajectories = data.frame(
squared_speed = c(160000,360000,640000),
distance = c(38098, 85692, 152220)
)
model = lm(squared_speed ~ distance, data = trajectories)
print(model)
Output:

$ Rscript speed_distance.r
Call:
lm(formula = squared_speed ~ distance, data = trajectories)
Coefficients:
(Intercept) distance
-317.708 4.206
Therefore the relationship between the squared velocity and the distance is predicted by the
regression to be:

2
v = 4.206 * d - 317.708.

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