A police car waits in hiding slightly off the highway. A...

A police car waits in hiding slightly off the highway. A speeding car is spotted by the police car doing $40 \mathrm{m} / \mathrm{s}$. At the instant the speeding car passes the police car, the police car accelerates from rest at $4 \mathrm{m} / \mathrm{s}^{2}$ to catch the speeding car. How long does it take the police car to catch the speeding car?

Key Concepts

Uniform Acceleration

This concept involves an object’s acceleration remaining constant over time, which allows the use of the kinematic equation relating displacement, time, and acceleration (s = 1/2at²) when the object starts from rest. It is fundamental in analyzing motions where the rate of change of velocity does not vary.

Constant Velocity Motion

When an object moves at a constant velocity, it covers equal distances in equal intervals of time. This linear motion is described by the equation s = vt, making it straightforward to compare and analyze relative movements against objects that are accelerating.

Relative Motion

Relative motion concerns the analysis of two or more objects moving in relation to each other. Understanding relative motion is essential in problems where one object needs to catch up to or overtake another, requiring the comparison of their respective position-time functions.

Equations of Motion

These equations describe an object's position, velocity, and acceleration over time. They are especially useful in solving problems that involve uniformly accelerated motion or constant velocity, enabling the prediction of future positions or times of intersection between moving objects.

Solving Quadratic Equations

Often in kinematics, setting two motion equations equal to each other to determine the meeting point results in a quadratic equation. Proficiency in solving quadratic equations, including identifying meaningful physical roots, is crucial for analyzing scenarios where the timing of events, such as catching up, is determined.

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