Two runners At noon (t=0), Alicia starts running along a...

Two runners At noon $(t=0),$ Alicia starts running along a long straight road at $4 \mathrm{mi} / \mathrm{hr}$. Her velocity decreases according to the function $v(t)=4 /(t+1),$ for $t \geq 0 .$ At noon, Boris also starts running along the same road with a 2 -mi head start on Alicia; his velocity is given by $u(t)=2 /(t+1),$ for $t \geq 0.$ a. Find the position functions for Alicia and Boris, where $s=0$ corresponds to Alicia's starting point. b. When, if ever, does Alicia overtake Boris?

Two runners At noon $(t=0),$ Alicia starts running along a long straight road at $4 \mathrm{mi} / \mathrm{hr}$. Her velocity decreases according to the function $v(t)=4 /(t+1),$ for $t \geq 0 .$ At noon, Boris also starts running along the same road with a 2 -mi head start on Alicia; his velocity is given by $u(t)=2 /(t+1),$ for $t \geq 0.$
a. Find the position functions for Alicia and Boris, where $s=0$ corresponds to Alicia's starting point.
b. When, if ever, does Alicia overtake Boris?

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Key Concepts

Integration of Velocity Functions

In kinematics, the position of an object is determined as the integral of its velocity function with respect to time. This process involves summing up the infinitesimal displacements over a time interval, and it provides a way to find the displacement or position function when the velocity function is known.

Initial Conditions

When integrating a velocity function to find a position function, an arbitrary constant of integration arises. Initial conditions, such as an object's starting position, are used to determine this constant, thereby providing a unique position function that correctly represents the motion over time.

Relative Motion and Overtaking

The concept of relative motion involves comparing the displacements or positions of two moving objects to determine when one overtakes another. By equating their position functions, one can solve for the time at which their positions are equal, indicating the point of overtaking.

Logarithmic Integration

Integrating functions of the form 1/(t+1) typically results in a logarithmic function, as the antiderivative is the natural logarithm of the absolute value of the argument. This is an important technique when working with velocity functions that vary inversely with time.

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