Kinematic Quantities

Fundamental physical quantities to describe position, change of position, and the temporal evolution of motion.

Conceptual context

Kinematics is the branch of physics that describes the motion of objects without considering the causes that produce it. To achieve this precise description, it relies on a set of fundamental quantities that capture where an object is, how fast it moves and how that velocity changes with time. These quantities, position, velocity and acceleration, form a chain of successive derivatives with respect to time, while the displacement is the net change of position between two instants.

Mastering their definitions and mutual relations is the first step to analysing any physical phenomenon, from the displacement of a vehicle to the movement of an electron. A common mistake is to confuse the distance traveled with the displacement, or the speed with the velocity. The distinction lies in the fact that the distance traveled and the speed are scalar quantities, whereas the displacement and the velocity are vectors. Understanding this difference prevents critical failures in problems involving closed paths or multidimensional movements.

🟢 Essential level

Imagine an everyday journey. Your position, formally the position vector running from the origin of the reference frame to your location, indicates where you are at a given instant within that frame. If you leave home, go round the block and end up at the pharmacy, the whole route you followed belongs to the idea of trajectory. The total length of that route is the distance traveled. By contrast, the straight arrow joining the starting point and the arrival point represents the displacement.

This distinction is the central idea of this module. Two movements can end at the same final point and yet have involved a very different distance traveled. It can also happen that you cover a lot of ground while the net displacement is small, or even zero, if you return almost to the starting point. For that reason, one cannot simply ask how far something has moved without specifying which quantity is being measured.

The first real physical skill here is conceptual: to distinguish what describes the actual route from what describes the net change between two positions. If this distinction fails, the quantities based on time built on top of it will also be misinterpreted.

[!TIP] Quick rule: the trajectory records the actual route followed, while the displacement only compares the initial point with the final one.

🔵 Formal level

Basic formulas

Motion is described through relations of change. The average velocity, the average speed and the average acceleration are the tools for summarising a journey. The first compares the displacement with the time, the second compares the total length of the path with the time, and the third compares the change in velocity with the time.

Average change relations. We have formulas to describe the overall behaviour of a moving body over a time interval:

1. Definition of displacement: It relates the initial position with the final position:

Displacement definition

2. Average velocity: It is the quotient of the net displacement and the elapsed time:

Average velocity

3. Average speed: It uses the total distance traveled covered along the path:

Average speed

4. Average acceleration: It quantifies how the velocity has varied over a given interval. To do so, we first calculate the change in velocity between the initial velocity and the final velocity:

Change in velocity definition

Average acceleration

Real student questions

• Can the velocity be negative? Yes. The sign indicates the direction relative to the axis. A negative value means motion in the direction opposite to the positive axis.
• Does zero acceleration mean rest? No. It means that the velocity does not change. A car travelling in a straight line at constant speed has zero acceleration, but it is not stopped.
• Why are two initial conditions needed? Because the acceleration is the second derivative of the position. The initial position and the initial velocity are required to fix the constants.

Final synthesis

The kinematic quantities (position, displacement, velocity and acceleration) form a hierarchy that encodes motion. Distinguishing between distance traveled and displacement prevents errors and connects geometry with energy. If the acceleration is known, two initial conditions are enough to reconstruct the trajectory.