Module 1: Describing Motion (Kinematics)
Kinematics is the language we use to describe how things move. We’ll learn about position, velocity, and acceleration – the fundamental building blocks for analysing motion in one, two, and three dimensions.
Position, Displacement, and Vectors
Position (r or x, y, z): An object’s location in space relative to a reference point (the origin). It requires both a distance and a direction from the origin. Because direction matters, position is a vector quantity. In 1D, we might just use x; in 2D or 3D, we use coordinates like (x, y) or (x, y, z). Units: meters (m).
Displacement (Δr or Δx): The change in an object’s position. It’s a vector pointing directly from the initial position to the final position. Calculated as Δr = r_final – r_initial. Importantly, displacement is not the same as the total distance traveled if the path isn’t straight. Units: meters (m).
Vectors vs. Scalars: Physics deals with two main types of quantities. Scalars have only magnitude (size), like distance, time, speed, mass, or temperature. Vectors have both magnitude and direction, like displacement, velocity, acceleration, or force. We often represent vectors visually as arrows (length indicates magnitude, arrowhead indicates direction).
Speed and Velocity
Velocity (v): How fast an object’s position is changing. Since position is a vector, velocity is also a vector – it has both speed and direction. It’s the rate of change of position. Units: meters per second (m/s).
Speed: The magnitude (size) of the velocity vector. It tells you how fast you’re going, but not which way. Speed is a scalar. Units: m/s.
Average Velocity (v_avg): The total displacement divided by the total time interval: v_avg = Δr / Δt. It depends only on the start and end points, not the path taken in between.
Instantaneous Velocity (v): The velocity at a specific moment in time. It’s what the speedometer (plus a compass) would read. Mathematically, it’s the derivative of position with respect to time: v = dr/dt.
Acceleration
Acceleration: How fast an object’s velocity is changing. Since velocity is a vector, acceleration is also a vector. Units: meters per second squared (m/s²).
Causes of Acceleration: You accelerate if you:
Speed up (magnitude of v increases).
Slow down (magnitude of v decreases – sometimes called deceleration).
Change direction (even if speed is constant, like going around a curve).
If acceleration a is in the same direction as velocity v, the object speeds up. If a is opposite to v, it slows down. If a is perpendicular to v, the direction of motion changes (like in circular motion).
Average vs. Instantaneous Acceleration: Similar to velocity, average acceleration is a_avg = Δv / Δt, while instantaneous acceleration is the derivative: a = dv/dt.
Kinematic Equations (Constant Acceleration)
Condition: These equations are shortcuts that ONLY work when acceleration a is constant (both magnitude and direction).
The Equations: They relate five key quantities: displacement (Δx or Δy), initial velocity (v₀), final velocity (v), acceleration (a), and time (t).
v = v₀ + at (Velocity from acceleration and time)
Δx = v₀t + ½at² (Displacement from initial velocity, acceleration, and time)
v² = v₀² + 2aΔx (Final velocity from initial velocity, acceleration, and displacement – time independent)
v² = v₀² + 2aΔx (Final velocity from initial velocity, acceleration, and displacement – time independent)
Δx = ½(v₀ + v)t (Displacement from average velocity and time)
Usage: If you know any three of the five quantities (and know a is constant), you can use these equations to find the other two.
Projectile Motion
Definition: The motion of an object launched into the air that moves only under the influence of gravity (ignoring air resistance).
Key Principle: Horizontal (x) and vertical (y) components of the motion are independent. You can analyze them separately.
Horizontal Motion: Constant velocity (since gravity only acts vertically). a_x = 0, so v_x = v₀x = v₀cos(θ) remains constant. Δx = v_x * t.
Vertical Motion: Constant downward acceleration a_y = -g (where g ≈ 9.8 m/s² near Earth). Use the 1D kinematic equations for the y-component: v_y = v₀y + a_yt, Δy = v₀yt + ½a_y*t², etc., where v₀y = v₀sin(θ).
Trajectory: The combination of constant horizontal velocity and uniformly accelerated vertical motion results in a parabolic path.
Module 2: Forces and Newton’s Laws
Introduction: Now we ask why things move (or stay still). The answer lies in forces – pushes and pulls – governed by Newton’s three fundamental laws of motion.
2.1 What is a Force?
Force (F): A push or a pull acting on an object. Forces can cause objects to accelerate (change their velocity). Force is a vector quantity, having both magnitude and direction. Units: Newtons (N). (1 N = 1 kg·m/s²).
Net Force (F_net or ΣF): The vector sum of all individual forces acting on an object. It’s the overall effective force.
2.2 Newton’s First Law (Inertia):
The Law: An object at rest stays at rest, and an object in motion stays in motion with the same velocity (constant speed and direction), unless acted upon by a non-zero net force.
Inertia: This property of objects to resist changes in their state of motion. Mass is a quantitative measure of inertia – more mass means more resistance to changes in velocity.
Key Idea: No net force means no acceleration (F_net = 0 implies a = 0). An object doesn’t need a force to keep moving at constant velocity, only to change its velocity.
2.3 Newton’s Second Law (F=ma):
The Law: The acceleration (a) of an object is directly proportional to the net force (F_net) acting on it and inversely proportional to its mass (m).
Equation: F_net = ma (or ΣF = ma). This is the central equation of classical mechanics.
Vector Nature: This is a vector equation. The acceleration vector a is always in the same direction as the net force vector F_net. We often apply it component-wise: ΣF_x = ma_x and ΣF_y = ma_y.
2.4 Newton’s Third Law (Action-Reaction):
The Law: For every action (force), there is an equal and opposite reaction (force).
Explanation: If object A exerts a force on object B (F_AB), then object B simultaneously exerts a force on object A (F_BA) that is equal in magnitude and opposite in direction: F_AB = -F_BA.
Important: The action and reaction forces always act on different objects. They never cancel each other out on a single object. (Example: You push on a wall; the wall pushes back on you).
2.5 Common Forces & Free Body Diagrams:
Weight (W or F_g): The force of gravity acting on an object. Near Earth’s surface, W = mg, directed downwards. g is the acceleration due to gravity (≈ 9.8 m/s²).
Normal Force (N or F_N): The perpendicular contact force exerted by a surface on an object pressing against it. It prevents the object from falling through the surface. Its magnitude adjusts as needed (up to a breaking point).
Friction (f or F_f): A contact force that opposes motion or attempted motion between surfaces.
Static Friction (f_s): Prevents motion from starting. Adjusts up to a maximum value: f_s ≤ μ_s N, where μ_s is the coefficient of static friction.
Kinetic Friction (f_k): Opposes motion once it has started. Usually constant: f_k = μ_k N, where μ_k is the coefficient of kinetic friction. Typically μ_k < μ_s.
Tension (T or F_T): The pulling force transmitted through a string, rope, cable, or wire when it’s pulled tight by forces acting from opposite ends. Acts along the direction of the rope.
Free-Body Diagram (FBD): A diagram showing one object isolated, with all the external forces acting on that object represented as vector arrows originating from the object. Essential first step for applying Newton’s Second Law.
2.6 Applying Newton’s Laws (Problem Solving):
Strategy:
Identify the object(s) of interest.
Draw a clear FBD for each object.
Choose a suitable coordinate system (often align one axis with acceleration).
Resolve all forces into x and y components according to your chosen axes.
Apply Newton’s Second Law separately for each component and each object: ΣF_x = ma_x, ΣF_y = ma_y.
Solve the resulting system of algebraic equations for the unknown quantities (e.g., acceleration, tension, normal force).
Constraints: Look for relationships between objects (e.g., connected objects often have the same magnitude of acceleration).