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Displacement is defined as the overall change in position of an object, and it's a vector quantity that has both magnitude and direction. Unlike distance, which is a scalar quantity and only measures the length of the path taken, displacement is concerned with the initial and final positions regardless of the path taken. In our example, the football quarterback's displacements during the three intervals were 15.0 m forward, -3.00 m backward, and 21.0 m forward again.

To grasp the concept of displacement better, imagine drawing a straight line from the start to the end point of the quarterback's movement. The length of this line represents the magnitude, and the arrow indicates the direction. Displacements can cancel each other out when they are in opposite directions, which is crucial when calculating the total displacement during different intervals of motion.

A time interval is the difference between two points in time and is critical when analyzing motion. Specifically, in kinematic problems, we are often interested in the elapsed time during which a particular motion or displacement occurs. In our example, each of the quarterback's movements occurs over specific time intervals: 2.50 s for the first run, 1.75 s for the push backward, and 5.20 s for the final forward run.

Understanding time intervals allows us to analyze each segment of motion individually before considering the motion as a whole. It's important to accurately measure these intervals to ensure the correct calculation of average velocities and other kinematic quantities.

Velocity calculation is fundamental to understanding motion. It's used to determine the rate of change of displacement with respect to time. Average velocity, in particular, can be calculated using the formula \( v_{\text{avg}} = \frac{\Delta x}{\Delta t} \) where \( \Delta x \) represents displacement and \( \Delta t \) denotes the time interval. For our quarterback, the average velocities for each interval are different because both the displacements and time intervals vary.

Remember that velocity, like displacement, is a vector and can be negative, indicating movement opposite to the chosen positive direction. This is why in our example, during the second interval when the quarterback is pushed backward, we use a negative displacement value to calculate the average velocity.

Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces causing the motion. The key elements of kinematics include displacement, velocity, acceleration, and time. It uses equations, also known as kinematic equations, to describe the motion of an object. Our quarterback's motion is a kinematic problem where we analyze his movement over time. By combining the concepts of displacement, time interval, and velocity calculation, we can describe and predict the quarterback's position at any time during the game.

By focusing on these concepts, you can understand kinematic problems better and apply the same principles to various real-world motion scenarios. Always be mindful of the difference between vector and scalar quantities, as it will guide you to solve these problems correctly.