Question

105–106. {Use of Tech} Races The velocity function and initial position of Runners A and B are given. Analyze the race that results by graphing the position functions of the runners and finding the time and positions (if any) at which they first pass each other.

A : v(t) = sin t; s(0) = 0 B. V(t) = cos t; S(0) = 0

Accepted Answer

Step 1: Understand the problem. You are given the velocity functions of two runners, A and B, as v_A(t) = sin(t) and v_B(t) = cos(t), along with their initial positions s_A(0) = 0 and s_B(0) = 0. The goal is to analyze the race by graphing their position functions and determining when and where they first pass each other. Step 2: Recall that the position function s(t) is the integral of the velocity function v(t). For Runner A, integrate v_A(t) = sin(t) to find s_A(t). Similarly, for Runner B, integrate v_B(t) = cos(t) to find s_B(t). Use the initial conditions s_A(0) = 0 and s_B(0) = 0 to determine the constants of integration. Step 3: Perform the integration. For Runner A, integrate sin(t) to get s_A(t) = -cos(t) + C_A. Use the initial condition s_A(0) = 0 to solve for C_A. For Runner B, integrate cos(t) to get s_B(t) = sin(t) + C_B. Use the initial condition s_B(0) = 0 to solve for C_B. Step 4: Graph the position functions s_A(t) and s_B(t) over a suitable interval of time. This will help visualize the motion of the runners and identify any points where their positions intersect. Step 5: Solve for the time t at which the runners first pass each other by setting s_A(t) = s_B(t). This involves solving the equation -cos(t) = sin(t). Once you find the time t, substitute it back into either position function to find the corresponding position where they pass each other.

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Key Concepts

Velocity and Position Functions

Integration

Graphing Functions