You drive \(4600 \mathrm{km}\) from coast to coast of the United States at \(65 \mathrm{mi} / \mathrm{h}(105 \mathrm{km} / \mathrm{h}),\) stopping an average of 30 min for rest after every 2 h of driving. (a) What's your average velocity for the entire trip? (b) How long does the trip take?

Kinematics is the branch of mechanics that deals with the motion of objects without considering the causes of the motion — forces and masses. When we think about kinematics, we concentrate on parameters such as displacement, velocity, and acceleration. In our exercise, we examine the motion of a cross-country trip, set to understand how fast and how long it takes to complete the journey.

Displacement in kinematics refers to the change in position of an object. It is a vector quantity, meaning it has both magnitude and direction. For our exercise, displacement is the total distance from coast to coast of the United States, which is given as 4600 km. We use this distance as the displacement in our average velocity calculation since the question assumes a straight path of travel from one coast to the other.

Under the broad umbrella of physics, the physics of motion, or dynamics, studies how and why objects move. While kinematics focuses only on the description of motion, dynamics also looks into the forces that cause or change this motion. In our average velocity problem, we deal with the concept of 'speed' and 'time.'

Speed, a scalar quantity, is the rate at which an object covers distance, and it's distinct from velocity, which is a vector quantity that stipulates the rate of change of the object's position. In the exercise, a consistent speed of 105 km/h is maintained while driving. Understanding these differences is key to mastering the concepts of the physics of motion and applying them to real-world problems like calculating travel times and average velocity.

Velocity time formulas are mathematical tools we use to quantify motion in physics. One of the most fundamental concepts is average velocity, which is described as the total displacement divided by the total time taken for the journey. The formula for average velocity is thus: \[\begin{equation}V_{avg} = \frac{displacement}{Total\ Time}\end{equation}\]In our problem, this formula helps us to calculate the average velocity of the entire trip. The trip includes periods of driving at a known speed and resting, influencing the total time. The key takeaway from the exercise is that average velocity takes into account the total time, including both motion and rest periods, providing a realistic measure of how fast the trip was completed from start to end.

To find the average velocity effectively, one must deal with each component – the driving and the rest – separately, then combine them. The driving time, resting breaks, and their duration are all variables that affect total time and thus the average velocity. By mastering velocity time formulas, students can solve complex motion problems in a step-by-step and logical manner.