Understanding Motion: Kinematics Fundamentals in Physics

The Fundamentals of Motion

Motion is defined as a change in position of a body relative to a reference point. Motion can be determined in two ways:

• By analyzing the trajectory and the relationship between position and time.
• By using the position vector as a function of time.

Concepts for the Study of Kinematics

• Mobile Object: The object in motion.
• Trajectory: The path followed by the mobile object.
• Origin/Reference Point: The point used as a reference to determine the mobile object's position.
• Position: The location of the mobile object relative to the origin.
• Displacement (Δs = s_f - s_i): The change in position of a mobile object between two instants of movement. It is the final position minus the initial position.
• Distance Traveled (e): The total length of the path traveled by the mobile object. If the object does not retrace its path, this value can be equal to the magnitude of the displacement.
• Velocity Vector: A vector quantity whose magnitude is the distance covered per unit time (speed), its direction is tangent to the trajectory, its sense is indicated by the arrowhead pointing towards the mobile object, and its point of application is the mobile object itself.
• Average Speed: Calculated by dividing the total distance traversed by a mobile object by the total time taken to traverse it.
• Instantaneous Velocity: The velocity of the mobile object at any given instant. The instantaneous velocity vector is always tangent to the path.

The position-time relationship can be expressed in three ways:

• Graphically: Position is always represented on the ordinate (y-axis), where the slope at each point corresponds to the velocity.
• Equation of Motion: The mathematical equation that relates position over time.
• Literary Description.

Types of Motion

Types of Motion Based on Speed Variation

• Uniform: If the position-time graph is a straight line.
• Varied: If the position-time graph is a curve.

Types of Motion According to the Direction of the Velocity Vector:

• Rectilinear: If the direction of the velocity vector remains constant.
• Curvilinear: If the direction of the velocity vector changes.

Uniform Rectilinear Motion (URM)

This motion has a straight path and a constant velocity magnitude. Its equation of motion is a first-degree polynomial.

s = vt + s₀

Here, s₀ is the initial position (the intercept), and v is the velocity (the slope). Its position-time graph (s-t) is a straight line.

Acceleration

Whenever the velocity changes, the motion is accelerated. Its magnitude (speed) or direction, or both, can change.

Acceleration can be calculated in two main cases:

• In rectilinear motion, where only its magnitude changes. Acceleration measures the change in the magnitude of velocity per unit time.

• In uniform circular motion, where only its direction changes. Acceleration measures the change in the direction of the velocity per unit time.

Accelerated Motion

Uniformly Accelerated Rectilinear Motion (UARM)

Its path is straight, and the magnitude of its velocity changes uniformly, resulting in constant acceleration.

• Equation of Motion: Its graph is a parabola.
• Equation of Velocity: v = v₀ + at. Its graph is a straight line whose slope represents the acceleration.
• Acceleration Equation: a = constant. Its graph is a line parallel to the time axis.
• Relationship between Position and Velocity: 2a(s - s₀) = v² - v₀²

Free Fall

This is the motion of a falling body subject only to the acceleration of gravity. On Earth, this ideal motion occurs only in a vacuum. It is a uniformly accelerated rectilinear motion with the acceleration of gravity, g = 9.8 m/s². For calculations, g is often taken as negative because it is directed towards the center of the Earth.

v = v₀ - 9.8t

s = s₀ + v₀t - ½ * 9.8 * t²

Uniform Circular Motion (UCM)

This motion has a circular path and a constant speed magnitude. Its characteristics can be described using linear or angular quantities.

• Linear Quantities:
  • Equation of Motion: s = s₀ + vt. Its graph is a straight line whose slope represents the speed.
  • Equation of Speed: v = constant. Its graph is a line parallel to the time axis.

Angular Quantities

• Angular Position (θ): The central angle swept by the radius vector to the mobile object when it is at position s. In uniform circular motion: θ = θ₀ + ωt. The graphical representation of θ-t is a straight line with slope ω.
• Angular Velocity (ω): The angle swept per second. Its value is constant in UCM.

Relationship Between Linear and Angular Quantities

• Relationship between Linear and Angular Positions: s = θR.
• Relationship between Linear and Angular Velocity: v = ωR.