AS Level Physics 9702: Kinematics Study Notes

Welcome to Kinematics!

Hello future Physicist! Kinematics might sound complicated, but it is simply the study of how things move—how fast they go, how far they travel, and how their motion changes, *without* worrying about the forces that cause the motion (that comes next in Dynamics!).

This chapter is the absolute foundation of mechanics. Mastering these concepts will make everything else in AS Physics much easier. Don't worry if this seems tricky at first; we'll break down the concepts using simple language and everyday examples!

1. Defining Key Terms: Scalars and Vectors

Before we jump into equations, we need to understand the language of motion. Physics separates quantities into two types:

1.1 Scalars vs. Vectors (A Quick Review)

Scalars: Quantities described only by their magnitude (size). They don't have a direction.
Examples: Distance, Speed, Mass, Time, Energy.

Vectors: Quantities described by both their magnitude and their direction. Direction matters!
Examples: Displacement, Velocity, Acceleration, Force, Momentum.

Memory Aid: A Very Vector has Very important Values (magnitude and direction).

1.2 Motion Definitions (Syllabus 2.1.1)
  • Distance (\(d\)): The total path length travelled. (Scalar)
    Example: If you walk 5 m East, and then 3 m West, the total distance is 8 m.
  • Displacement (\(s\)): The shortest straight-line distance from the starting point to the final point, including direction. (Vector)
    Example: In the walk above, your final displacement is 2 m East.
  • Speed: Rate of change of distance.
    $$ \text{Speed} = \frac{\text{Distance}}{\text{Time}} $$ (Scalar. Unit: \(\text{m s}^{-1}\))
  • Velocity (\(v\)): Rate of change of displacement.
    $$ \text{Velocity} = \frac{\text{Displacement}}{\text{Time}} $$ (Vector. Unit: \(\text{m s}^{-1}\))
  • Acceleration (\(a\)): Rate of change of velocity.
    $$ a = \frac{\text{Change in Velocity}}{\text{Time taken}} = \frac{v - u}{t} $$ (Vector. Unit: \(\text{m s}^{-2}\))

Key Takeaway: Always be precise! Speed and distance don't care about direction; velocity and displacement absolutely do. When calculating velocity, you must use displacement, not distance.

2. Graphical Methods for Motion (Syllabus 2.1.2 - 2.1.5)

Graphs are essential tools in Kinematics. They allow us to instantly understand the motion of an object and calculate key quantities using just the gradient or the area.

2.1 Displacement-Time (d-t) Graphs

These graphs plot displacement (\(s\)) on the y-axis against time (\(t\)) on the x-axis.

  • What the Gradient Represents: The velocity (Syllabus 2.1.4). $$ \text{Gradient} = \frac{\text{Change in } y}{\text{Change in } x} = \frac{\Delta s}{\Delta t} = \text{Velocity} $$
  • Interpreting Shapes:
    • A horizontal line (zero gradient) means zero velocity (the object is stationary).
    • A straight line with a constant positive gradient means constant positive velocity (uniform motion).
    • A curve means the velocity is changing, which means the object is accelerating.
2.2 Velocity-Time (v-t) Graphs

These graphs plot velocity (\(v\)) on the y-axis against time (\(t\)) on the x-axis. These are the most powerful graphs for kinematic calculations.

  • What the Gradient Represents: The acceleration (Syllabus 2.1.5). $$ \text{Gradient} = \frac{\text{Change in } y}{\text{Change in } x} = \frac{\Delta v}{\Delta t} = \text{Acceleration} $$
  • What the Area Represents: The displacement (Syllabus 2.1.3).
    The area is found by multiplying \(v \times t\), and since \(v = s/t\), then \(s = v \times t\).
Step-by-Step: Analysing a Velocity-Time Graph

Imagine a cyclist speeding up, cruising, and then braking:

  1. Initial Segment (Speeding Up): The line slopes upwards.
    Gradient: Positive and constant. This means constant acceleration.
  2. Middle Segment (Cruising): The line is horizontal.
    Gradient: Zero. This means zero acceleration (constant velocity).
  3. Final Segment (Braking): The line slopes downwards.
    Gradient: Negative and constant. This means constant negative acceleration (or constant deceleration).
  4. Total Displacement: Calculate the area of the shape under the line (usually a trapezoid or a combination of rectangles and triangles).

Quick Review: Graph Rules

  • d-t graph: Gradient gives Velocity.
  • v-t graph: Gradient gives Acceleration; Area gives Sinplacement (Displacement).

3. Uniformly Accelerated Motion (SUVAT Equations) (Syllabus 2.1.6, 2.1.7)

Many physics problems involve motion where the acceleration is constant (uniform) and the object moves in a straight line. For these specific conditions, we use a set of powerful mathematical tools often called the SUVAT equations.

3.1 The Variables

We use five key variables, all of which are vectors (meaning we must choose a positive direction and stick to it!):

  • S = Displacement (\(s\))
  • U = Initial velocity (\(u\))
  • V = Final velocity (\(v\))
  • A = Constant acceleration (\(a\))
  • T = Time taken (\(t\))
3.2 The Four SUVAT Equations

These equations are derived from the definitions of velocity and acceleration, and from the properties of the velocity-time graph for constant acceleration.

  1. $$ v = u + at $$ (The definition of acceleration rearranged)
  2. $$ s = \frac{(u+v)}{2} t $$ (Displacement = Average velocity \(\times\) time)
  3. $$ s = ut + \frac{1}{2} a t^2 $$ (Derived by substituting (1) into (2))
  4. $$ v^2 = u^2 + 2as $$ (Derived by eliminating \(t\) from (1) and (2))

Important Tip for Using SUVAT:
To solve any kinematics problem, you must know three of the five variables to find a fourth. Write down the SUVAT list, fill in what you know, and identify what you need to find.

Common Mistake to Avoid:
You can only use the SUVAT equations if the acceleration, \(a\), is constant. If acceleration changes (e.g., if air resistance is present and changing speed), you must rely on graphical methods or calculus (though calculus is generally beyond the AS level scope).

*Derivation Spotlight (Syllabus 2.1.6)*

It is important to know how the equations are derived from the basic definitions:

  1. Deriving \(v = u + at\):
    By definition, acceleration is the change in velocity divided by time: $$ a = \frac{v - u}{t} $$ Rearranging gives: $$ at = v - u $$ $$ \mathbf{v = u + at} $$
  2. Deriving \(s = ut + \frac{1}{2} a t^2\):
    We know displacement is the average velocity multiplied by time: $$ s = \left(\frac{u+v}{2}\right) t $$ Substitute Equation 1 (\(v = u + at\)) into this expression: $$ s = \left(\frac{u + (u + at)}{2}\right) t $$ $$ s = \left(\frac{2u + at}{2}\right) t $$ $$ s = \left( u + \frac{1}{2} at \right) t $$ $$ \mathbf{s = ut + \frac{1}{2} a t^2} $$

Key Takeaway: The SUVAT equations are your mathematical workhorses for any motion with constant acceleration in a straight line.

4. Free Fall and Acceleration of Free Fall (g) (Syllabus 2.1.7, 2.1.8)

A crucial application of uniform acceleration is the motion of objects under gravity, assuming no air resistance. This is called free fall.

4.1 The Acceleration of Free Fall

When an object falls solely due to gravity in a vacuum, its acceleration is constant. This value is known as the acceleration of free fall (\(g\)).

  • Near the Earth's surface, the accepted value of \(g\) is approximately \(9.81 \text{ m s}^{-2}\).
  • In these problems, you simply substitute \(a = g\) into the SUVAT equations.
  • Crucial Rule: Direction is Key! When using SUVAT for vertical motion:
    • Choose one direction (usually upwards or downwards) as positive.
    • If you choose UP as positive, then \(g\) must be negative (\(a = -9.81 \text{ m s}^{-2}\)).
    • If you choose DOWN as positive, then \(g\) is positive (\(a = +9.81 \text{ m s}^{-2}\)).
Did you know?

Galileo Galilei famously demonstrated that in the absence of air resistance, all objects fall at the same rate, regardless of their mass. The feather and the hammer fall together on the Moon!

4.2 Experiment to Determine g (Syllabus 2.1.8)

A common experiment uses an object falling vertically, timed electronically.

  1. A small object (like a ball bearing) is held by an electromagnet and starts falling when the current is switched off. This sets the timer (\(t=0\)).
  2. The ball falls through two or more light gates placed at measured vertical distances (\(s\)).
  3. The time taken (\(t\)) to pass between the gates is recorded electronically.
  4. Since the object starts from rest, the initial velocity \(u=0\). We can use the SUVAT equation: $$ s = ut + \frac{1}{2} a t^2 $$ Since \(u=0\): $$ s = \frac{1}{2} g t^2 $$
  5. By rearranging, \(g = \frac{2s}{t^2}\). If you plot a graph of \(s\) (y-axis) against \(t^2\) (x-axis), the graph should be a straight line through the origin with a gradient equal to \(\frac{1}{2} g\).

Key Takeaway: Free fall means the only acceleration acting is \(g\). Always define your positive direction when applying SUVAT.

5. Motion in Two Dimensions (Projectile Motion) (Syllabus 2.1.9)

Projectile motion describes the path of an object launched into the air (like a thrown ball or a bullet), assuming air resistance is negligible.

5.1 The Principle of Independence of Motion

The key to solving projectile problems is realizing that the horizontal motion and the vertical motion are completely independent of each other.

Think of two separate systems working simultaneously:

  1. Horizontal Motion (x-direction):
    • There are no forces acting horizontally (assuming zero air resistance).
    • Therefore, the acceleration is zero (\(a_x = 0\)).
    • The object moves with constant velocity (\(v_x\)).
    • Equation used: $$ \text{Displacement } s_x = v_x \times t $$
  2. Vertical Motion (y-direction):
    • The only force acting is gravity.
    • The acceleration is constant and equal to \(g\), acting downwards (\(a_y = g\)).
    • The object uses the SUVAT equations.
    • Equation used (e.g., if starting from rest vertically): $$ s_y = u_y t + \frac{1}{2} g t^2 $$
5.2 How to Solve Projectile Problems

The link between the two independent motions is time (\(t\)). The time it takes for a projectile to travel horizontally is exactly the same time it takes to travel vertically.

Step-by-Step Procedure:

  1. Resolve the Initial Velocity: If the object is launched at an angle \(\theta\), break the initial velocity \(u\) into horizontal (\(u_x\)) and vertical (\(u_y\)) components: $$ u_x = u \cos \theta $$ $$ u_y = u \sin \theta $$
  2. Horizontal Analysis: Use \(s_x = u_x t\) to find the horizontal range or time of flight.
  3. Vertical Analysis: Use the SUVAT equations with \(a_y = \pm g\) to find the maximum height, time to reach maximum height (where \(v_y = 0\)), or the time of flight.
  4. Combine: Often, you must find \(t\) using the vertical motion information, and then substitute that \(t\) back into the horizontal equation to find the range.

Analogy: Imagine throwing a ball horizontally off a cliff. The moment it leaves your hand, gravity starts pulling it down (vertical motion), but its horizontal speed remains unchanged (horizontal motion) until it hits the ground.

Key Takeaway: Projectile motion is simply two one-dimensional motions happening at once, connected by the variable time (\(t\)). Separate horizontal (\(a=0\)) and vertical (\(a=g\)) calculations.