Where do they meet? Kelly started at noon (t=0) riding a...

Where do they meet? Kelly started at noon $(t=0)$ riding a bike from Niwot to Berthoud, a distance of $20 \mathrm{km},$ 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. a. Make a graph of Kelly's distance from Niwot as a function of time. b. Make a graph of Sandy's distance from Berthoud as a function of time. c. How far has each person traveled when they meet? When do they meet? 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? e. Looking ahead: With the velocity functions given in part (d), make a conjecture about the maximum distance each person can ride (given unlimited time).

Where do they meet? Kelly started at noon $(t=0)$ riding a bike from Niwot to Berthoud, a distance of $20 \mathrm{km},$ 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.
a. Make a graph of Kelly's distance from Niwot as a function of time.
b. Make a graph of Sandy's distance from Berthoud as a function of time.
c. How far has each person traveled when they meet? When do they meet?
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?
e. Looking ahead: With the velocity functions given in part (d), make a conjecture about the maximum distance each person can ride (given unlimited time).

Key Concepts

Integration of Velocity Functions

This concept involves calculating the distance traveled as the integral of the velocity function over time. Since velocity represents the rate of change of position, integrating the velocity function yields a distance function, which is fundamental in analyzing motion and comparing positions of moving objects.

Distance-Time Graphs

Distance-time graphs are used to visually illustrate an object's position relative to a starting point as a function of time. They help in understanding how distance accumulates over time and provide a clear way to compare motions between different objects, particularly when determining meeting points.

Relative Motion and Meeting Points

Relative motion involves analyzing the movement of two or more objects with respect to each other. By equating their distance functions, one can determine when and where they meet. This concept is critical in problems where objects start from different locations and move toward or away from each other.

Parameter Conditions for Intersection

When velocity functions are generalized with parameters, determining whether the paths of two moving objects will intersect requires establishing relationships between these parameters and the distance between starting points. Such conditions ensure that the combined distance traveled by both objects can bridge the gap between their starting positions.

Asymptotic Behavior and Maximum Distance

This concept focuses on understanding the long-term behavior of distance functions, especially when velocity decreases over time. As time approaches infinity, the distance function may converge to a maximum value, representing the maximum distance that an object can travel. This analysis is valuable for predicting limits on travel in scenarios where speed decreases continuously with time.

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