Question

The smaller the curvature in a bend of a road, the faster a car can travel. Assume that the maximum speed around a turn is inversely proportional to the square root of the curvature. A car moving on the path \(y=\frac{1}{3} x^{3}(x\) and \(y\) are measured in miles) can safely go 30 miles per hour at \(\left(1, \frac{1}{3}\right)\). How fast can it go at \(\left(\frac{3}{2}, \frac{9}{8}\right) ?\)

Short answer

The car can go 390 miles per hour at the point \((3/2, 9/8)\)

Step-by-step solution

Determine the curvature

Curvature \(\kappa\) of a function is given by the formula: \[\kappa = \frac{|y''|}{(1 + (y')^{2})^{3/2}}\] where \(y'\) is the first derivative and \(y''\) is the second derivative of the function. Here the function is \(y=\frac{1}{3} x^{3}\), so first calculate \(y'\) and \(y''\).

Calculate the first and second derivatives

We have \(y'=x^{2}\) and \(y''=2x\). Now, substituting these values in the formula for curvature we have: \[\kappa = \frac{|2x|}{(1 + x^{4})^{3/2}}\]

Calculate the curvature at the first point

Substitute the first point \((1, 1/3)\) into the curvature formula to get \(\kappa_{1}\). \[\kappa_{1} = \frac{|2(1)|}{(1 + (1)^{4})^{3/2}} = \frac{2}{\sqrt{2}} = \sqrt{2}\]

Calculate the curvature at the second point

Similarly, substitute the second point \((3/2, 9/8)\) into the curvature formula to get \(\kappa_{2}\). \[\kappa_{2} = \frac{|2(3/2)|}{(1 + (3/2)^{4})^{3/2}} = \frac{3}{(1+\frac{81}{16})^{3/2}} = \frac{3}{\sqrt{117}}\]

Apply the proportionality to find the speed at the second point

Now according to the problem, speed is inversely proportional to the square root of the curvature, i.e., \(s = k/\sqrt{\kappa}\). Given that the car safely travels at 30 mph at the point (1, 1/3) where the curvature is \(\sqrt{2}\), we have \(30 = k/\sqrt{2}\). Solving for \(k\), we get \(k = 30\sqrt{2}\). Therefore, at the point \((3/2, 9/8)\) the speed \(s_{2}\) would be \(s_{2} = k/\sqrt{\kappa_{2}} = \frac{30\sqrt{2}}{\sqrt{\frac{3}{\sqrt{117}}}} = \frac{30\sqrt{2}\sqrt{117}}{3} = 390 mph\).