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?

Verified step by step guidance

1. 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.
2. 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.
3. 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 \(x_S(t)\).
4. 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.
5. 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 \(D\).

Key Concepts

Position as an Integral of Velocity

The position of each rider over time is found by integrating their velocity functions. Since velocity varies with time, integrating v(t) or u(t) from 0 to t gives the distance traveled from the starting point. This allows us to express each rider's location as a function of time, essential for determining when and where they meet.

Meeting Condition Based on Total Distance

The riders meet when the sum of the distances each has traveled equals the total distance D between the towns. Mathematically, this means the integral of v(t) plus the integral of u(t) from 0 to the meeting time t equals D. Understanding this condition helps set up an equation to solve for the meeting time.

Convergence of Improper Integrals and Parameter Constraints

To ensure the riders actually meet, the combined distances traveled over infinite time must be at least D. This involves analyzing the improper integrals of v(t) and u(t) as t approaches infinity. The parameters A, B, and D must satisfy conditions so that the total possible distance covered is at least D, guaranteeing the riders pass each other.