Formulas and online calculators exist to compute the 1/4 mile elapsed time from

Question

Formulas and online calculators exist to compute the 1/4 mile elapsed time from a car’s horsepower and weight. For example one formula that appears in some places is:

t = 6.290 · ( w )1/3 hp

where t is elapsed time in seconds and w is weight in pounds. Some things about this formula are plausible from physics. Assuming constant acceleration a = v/t, mass m, distance d = 1/4 mile, and force F = ma:

hp = F · v

= (m · v/t) · v (2d)2

= m· t ,velocity v = 2d/t, a = 2d/t2 t

= 4md2 t3

which finally leads to t = C · (w/hp)1/3 for some constant C. But the details of air resistance, transmission, friction, horsepower curve, and units make it complicated.

(a) Assume a model of the form

t = · (w/hp)
with unknown parameters (, ). Use the mtcars data set in R to fit the model with lm.

(b) Compare the fitted model with the one above.

(c) Predict the 1/4-mile time for the 2016 Honda Accord Sport Manual and get a prediction interval. Use a weight of 3400 lbs and horsepower of 189. Does your prediction interval contain the true value, or reasonable measures of it?

(d) Identify problems with residuals in the fitted model, report the coefficient of determination R2, and discuss whether it is reasonable to use data from the early 1970s for predictions today.

Explanation / Answer

a)

F = ma:

hp = F · v

= (m · v/t) · v (2d)2

= m· t ,velocity v = 2d/t, a = 2d/t2 t

= 4md2 t3

a) t = C · (w/hp)1/3

t = · (w/hp)

equating both

C · (w/hp)1/3= · (w/hp)

· =C · (w/hp)1/3/ (w/hp

· =c/3

C)t = 6.290 · ( w )1/3 hp

t=6.290*3400/(3*189)

d)sometimes it can be reasonable

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