Comprehensive Study Notes on Kinematics and Linear Motion

Fundamental Definitions of Kinematics: Trajectory, Displacement, and Distance

In the study of kinematics, it is essential to distinguish between the path an object takes and its change in position. Trajectory is defined as the set of all points occupied by a moving body over time, whereas displacement ($\Delta \vec{x}$) is a vector quantity representing the straight line connecting the starting point to the ending point. A common misconception is to equate the path traveled (the actual length of the trajectory) with displacement. In any movement that is not strictly in a single direction along a straight line, the path traveled ($d$) will always be greater than or equal to the magnitude of the displacement ($|\Delta \vec{x}|$). For any two points A and B, there are infinite possible trajectories, but only one displacement vector. In a closed trajectory where the object returns to its origin, the displacement is $\vec{0}$, even if the distance traveled is non-zero.

The path traveled is a scalar magnitude, meaning it is defined solely by its numerical value and unit, whereas displacement is a vector magnitude that requires a direction and sense. If a particle moves in a straight line without reversing direction, the trajectory, path traveled, and displacement magnitude are equal. However, if the motion occurs on a plane or involves curves, these values diverge. For example, if two people travel from Chillán to Concepción, one taking a winding old road and the other a straight new road, and both arrive in the same amount of time, their displacement is identical because the starting and ending points are the same, despite their different trajectories and average speeds.

Principles of Velocity and Speed

Average speed is a scalar quantity defined as the quotient between the distance (path) traveled and the total time taken ($v = d / t$). Conversely, average velocity ($\vec{v}$) is a vector quantity defined as the quotient between the displacement and the total time ($\vec{v} = \Delta \vec{x} / t$). In the International System (S.I.), both share the same units, typically meters per second ($m/s$), though other units like kilometers per hour ($km/h$) or meters per minute ($m/min$) are commonly used. If a particle has a null average velocity over a time interval $T$, it implies that there was no net displacement, meaning the trajectory was closed or the particle returned to the origin.

A constant velocity implies that the moving body maintains both a constant speed and a constant direction, which means it does not experience acceleration ($a = 0$). If an object moves with a constant speed of $40\,km/h$ but enters a roundabout, it is still accelerating because its direction (velocity vector) is changing. Similarly, velocity is considered to be changing if the object is braking (negative acceleration) or starting from rest (positive acceleration). In scenarios involving multiple stages of motion, the total average speed is not necessarily the arithmetic mean of the speeds. For instance, if an object travels half a distance at $10\,m/s$ and the other half at $40\,m/s$, the average speed for the entire trip is calculated using the harmonic mean formula, resulting in $16\,m/s$.

Acceleration and the Dynamics of Motion

Acceleration is the rate at which velocity changes over time. It is measured in units of length over time squared, such as $m/s^2$. An acceleration of $0.2\,m/s^2$ signifies that for every second that passes, the object's speed increases by $0.2\,m/s$. If an airplane starts from rest ($0\,m/s$) and reaches a speed of $100\,m/s$ in $10\,s$, the magnitude of its acceleration is calculated as the change in speed ($100 - 0$) divided by the time ($10$), resulting in $10\,m/s^2$. In a Uniform Rectilinear Motion (M.R.U.), the velocity remains constant, meaning the displacement varies linearly with time and the acceleration is zero.

Numerical problems often combine these concepts. Consider a vehicle moving at $40\,m/s$ that increases its speed to $52\,m/s$ over a period of $12\,s$. The calculated acceleration is $1\,m/s^2$. In encounter problems, where two objects move toward each other, the time to meet ($t_e$) is found by dividing the total distance by the sum of their speeds. For two objects 144 meters apart moving at $10\,m/s$ and $8\,m/s$, they would meet in $144 / (10 + 8) = 8\,s$. This can be represented graphically where the position-time lines for each object intersect at the 8-second mark.

Graphical Interpretation of Linear Motion

Position-time graphs ($x$ vs $t$) and velocity-time graphs are critical for understanding motion patterns. In an $x$ vs $t$ graph, the slope represent velocity. A horizontal line indicates the object is at rest. For example, if a player's position is graphed over $45\,s$, and the position remains constant between $t = 5\,s$ and $t = 10\,s$, the player is stationary during that interval. If a particle starts at the origin, moves to $10\,m$ in $2\,s$, stays there for $2\,s$, and then returns to the origin in $2\,s$, the average velocity for the entire trip is $0\,m/s$ because the final displacement is zero, even though the total distance covered is $20\,m$.

Consider a complex vector displacement problem: A man walks 4 blocks East, 3 North, 3 East, 6 South, and 3 West. To find the total displacement vector, we sum the components: East-West reaches $4 + 3 - 3 = 4$ blocks East, and North-South reaches $3 - 6 = -3$ blocks (3 blocks South). The module (magnitude) of the displacement is calculated via the Pythagorean theorem: $\sqrt{4^2 + (-3)^2} = 5$ blocks. In another scenario, an airplane flies $300\,km$ North and then $400\,km$ East in $2\,h$. The total distance is $700\,km$, making the average speed $350\,km/h$. The displacement is the hypotenuse of the triangle, $\sqrt{300^2 + 400^2} = 500\,km$, leading to a velocity magnitude of $250\,km/h$.

Analytical Problem Solving and Development

Specific motion scenarios require precise tracking of distances and positions. In a rectangular path where a particle starts at the origin and returns after traversing the perimeter, the total distance is the sum of all sides (e.g., if the rectangle has vertices at $(0,0), (6,0), (6,2), (0,2)$, the total distance for a full loop is $16\,m$), while the total displacement is $0\,m$. If a graph shows a particle moving from position $-5\,m$ to $15\,m$ and back, the distance traveled is the sum of the absolute segments, while displacement is simply the final position minus the initial position.

In comparative efficiency problems, such as a motorist traveling a route at $60\,km/h$ in $2\,h$, the distance is established as $120\,km$. If the speed increases to $90\,km/h$, the new time calculation is $120 / 90 = 1.33\,h$, or $80\,minutes$. Comparing this to the original $120\,minutes$, the time saved is $40\,minutes$. For a graph representing the position of two cars, the one with the steeper slope is the faster vehicle. For instance, if car 1 travels $100\,m$ in $10\,s$, its velocity is $10\,m/s$, whereas if car 2 travels $25\,m$ in $10\,s$, its velocity is $2.5\,m/s$. Such comparisons demonstrate that velocity is the physical quantity defining the rate of change of position in a specific direction.