Question

Where do they meet? Kelly started at noon (t=0) riding a bike from Niwot to Berthoud, a distance of 20 km, with velocity v(t) = 15 / (t + 1)² (decreasing because of fatigue). Sandy started at noon (t=0) riding a bike in the opposite direction from Berthoud to Niwot with velocity u(t) = 20 / (t + 1)² (also decreasing because of fatigue). Assume distance is measured in kilometers and time is measured in hours.

d. More generally, if the riders’ speeds are v(t)=A(t+1)² and u(t)=B(t+1)² and the distance between the towns is D, what conditions on A, B, and D must be met to ensure that the riders will pass each other?

Accepted Answer

First, understand that the riders start at opposite ends of a distance $$D$$ kilometers apart and move towards each other. Their positions as functions of time $$t (in $$hours) can be found by integrating their velocity functions, since velocity is the derivative of position with respect to time. Express the position of Kelly starting from Niwot as $$x_K(t) = \$$int_0$$^t v(s) \, ds$$, where $$v(t) = \frac{A}{(t+1)^2$}. $$Similarly, express Sandy's position starting from Berthoud as $$x_S(t) = D - \$$int_0$$^t u(s) \, ds$$, where $$u(t) = \frac{B}{(t+1)^2$}. $$Note that Sandy's position decreases from $$D$$ towards 0. Calculate the integrals for both riders: $$\$$int_0$$^t \frac{A}{(s+1)^2$} ds$$ and $$\$$int_0$$^t \frac{B}{(s+1)^2$} ds. $$Recall that $$\int \frac{1}{(s+1)^2$} ds = -\frac{1}{s+1} + C. $$Use this to find explicit expressions for $$x_K(t)$$ and $$x_S(t). $$Set the positions equal to find the meeting time $$t^*$$: $$x_K(t^*) = x_S(t^*). $$This gives an equation involving $$A$$, $$B$$, $$D$$, and $$t^*$$ that can be solved to find when and if they meet. To ensure the riders pass each other, the meeting time $$t^*$$ must be a positive real number. Analyze the equation to find the condition on $$A$$, $$B$$, and $$D$$ that guarantees such a $$t^*$$ exists. This typically means the combined distance they cover over time must be at least $$D.$$

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Key Concepts

Position as an Integral of Velocity

Meeting Condition Based on Total Distance

Convergence of Improper Integrals and Parameter Constraints