Question

Two runners At noon (t=0), Alicia starts running along a long straight road at 4 mi/hr. Her velocity decreases according to the function v(t) = 4 / t + 1 for t≥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≥0. Assume t is measured in hours.

b. When, if ever, does Alicia overtake Boris?

Accepted Answer

First, understand that to find when Alicia overtakes Boris, we need to find the time $$ t $$ when both have run the same distance from the starting point of Alicia. Since Boris has a 2-mile head start, his position at time $$ t $$ will be his initial 2 miles plus the distance he covers after time $$ t . $$Express the position functions for both runners by integrating their velocity functions over time. For Alicia, her position function $$ s_A(t)  is $$given by integrating $$ v(t) = \frac{4}{t+1} $$ from 0 to $$ t $$:

$$
 s_A(t) = \int_0^t \frac{4}{x+1} \, dx
$$ Similarly, for Boris, his position function $$ s_B(t) $$ includes his 2-mile head start plus the integral of his velocity function $$ u(t) = \frac{2}{t+1} $$ from 0 to $$ t $$:

$$
 s_B(t) = 2 + \int_0^t \frac{2}{x+1} \, dx
$$ Calculate both integrals to find explicit expressions for $$ s_A(t) $$ and $$ s_B(t) . $$These will involve natural logarithms because the integral of $$ \frac{1}{x+1}  is$$ $$ \ln|x+1| . $$Set the two position functions equal to each other to find the time when Alicia overtakes Boris:

$$
 s_A(t) = s_B(t)
$$

Solve this equation for $$ t  to $$find the time(s) when Alicia catches up to Boris.

Verified video answer for a similar problem:

Key Concepts

Velocity and Position Relationship

Solving Equations to Find Intersection Points

Handling Velocity Functions with Variable Denominators