(III) A fugitive tries to hop on a freight train traveling...

(III) A fugitive tries to hop on a freight train traveling at a constant speed of 5.0 m/s. Just as an empty box car passes him, the fugitive starts from rest and accelerates at $a =$ 1.4 m/s$^2$ to his maximum speed of 6.0 m/s, which he then maintains. ($a$) How long does it take him to catch up to the empty box car? ($b$) What is the distance traveled to reach the box car?

Key Concepts

Uniformly Accelerated Motion

This concept involves motion with a constant acceleration, where key equations, such as the displacement equation (?x = v?t + ½at²) and the velocity equation (v = v? + at), are used to describe how an object's velocity and position change over time. It is fundamental to analyze scenarios in which an object is speeding up or slowing down at a constant rate.

Constant Velocity Motion

In constant velocity motion, the speed and direction of the object remain unchanged over time. The displacement in such cases is simply the product of velocity and time (?x = vt). This concept simplifies the analysis of motion by removing any acceleration components, making it easier to predict future positions.

Relative Motion Analysis

Relative motion analysis involves comparing the motion of two objects by observing their relative speeds and positions. This technique is crucial when determining the conditions needed for one moving object to catch up with or meet another by considering the differences in their velocities and the timing of their movements.

Piecewise Kinematics Analysis

This approach breaks down a motion problem into distinct phases where different kinematic conditions apply—such as an acceleration phase followed by a constant speed phase. By analyzing each segment separately and then connecting the results, one can accurately determine overall quantities like travel time and total distance covered.

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