After rounding the final turn in the bell lap, two runners...

After rounding the final turn in the bell lap, two runners emerged ahead of the pack. When runner A is $200 \mathrm{ft}$ from the finish line, his speed is $22 \mathrm{ft} / \mathrm{sec}$, a speed that he maintains until he crosses the line. At that instant of time, runner $\mathrm{B}$, who is $20 \mathrm{ft}$ behind runner A and running at a speed of $20 \mathrm{ft} / \mathrm{sec}$, begins to sprint. Assuming that runner B sprints with a constant acceleration, what minimum acceleration will enable him to cross the finish line ahead of runner A?

Key Concepts

Quadratic Equation Application

In kinematics, setting up an equation that models the scenario often leads to a quadratic equation in the time variable. Solving this quadratic equation, or determining the conditions for a unique solution by setting the discriminant to zero, is a key step in finding the minimum required acceleration or similar parameters in problems involving accelerated motion.

Relative Motion Analysis

Relative motion involves examining the motion of one object as observed from another moving object. This concept is essential when comparing the progress of two entities moving simultaneously but with different speeds or accelerations. By analyzing motion relative to each other, one can determine the conditions necessary for one to overtake or finish ahead of the other.

Uniform Motion and Constant Speed

Uniform motion occurs when an object travels at a constant speed in a straight path without acceleration. The displacement in such cases can be calculated simply by multiplying the speed by the time elapsed. This concept is important in scenarios where an object’s motion does not change over time, which contrasts with accelerating objects.

Kinematic Equations

Kinematic equations provide mathematical relationships between displacement, initial velocity, acceleration, and time. In problems involving constant acceleration, these equations are crucial to determine how far an object travels or how long it takes to reach a certain point. They allow the formulation of equations that describe motion and enable the solving of variables when some of the others are known.

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