Welcome to Kinematics: Describing Motion in a Straight Line!
Hello future Mathematician! Welcome to the very first chapter of Mechanics (M1). Kinematics might sound like a complicated word, but it simply means the study of how things move, without worrying about why they move (that comes later!).
In this chapter, we focus entirely on particles moving along a straight line. This simplification allows us to build powerful mathematical tools that are used everywhere, from designing rollercoasters to calculating satellite orbits. Don't worry if this seems tricky at first—we will break down every concept step-by-step!
Section 1: The Foundations – Scalars vs. Vectors
The biggest hurdle in kinematics is understanding the difference between two types of quantities: Scalars and Vectors.
1.1 Key Definitions: Scalar Quantities
A Scalar quantity only has magnitude (size). It does not care about direction.
- Example: Temperature, Mass, Time, and most importantly for us: Distance and Speed.
1.2 Key Definitions: Vector Quantities
A Vector quantity has both magnitude (size) and direction. In M1, direction is usually defined simply as positive (+) or negative (-) along the straight line.
- Example: Force, Momentum, and most importantly for us: Displacement, Velocity, and Acceleration.
Memory Aid: Think of a Vampire—they always need a direction (where to bite next)! Things starting with V (Velocity, Vector) need direction.
1.3 Displacement and Distance (s)
Displacement (s) (Vector, Units: m):
This is the shortest distance from the starting point (the origin) to the final position. It includes direction.
- Example: If you walk 5 m East (+5 m) and then 2 m West (-2 m), your total Displacement is \(5 - 2 = 3\) m East.
Distance (Scalar, Units: m):
This is the total length traveled along the path.
- Example: Using the example above, your total Distance traveled is \(5 + 2 = 7\) m.
1.4 Velocity and Speed (v and u)
Velocity (v or u) (Vector, Units: \(\text{ms}^{-1}\)):
This is the rate of change of displacement.
\[\text{Velocity} = \frac{\text{Change in Displacement}}{\text{Time taken}}\]
Speed (Scalar, Units: \(\text{ms}^{-1}\)):
This is the rate of change of distance. Speed is the magnitude of velocity.
1.5 Acceleration (a)
Acceleration (a) (Vector, Units: \(\text{ms}^{-2}\)):
This is the rate of change of velocity. If an object is accelerating, its velocity is changing (getting faster or slower).
If acceleration is constant, we can use the simple formula:
\[a = \frac{v - u}{t}\]
Where \(u\) is the initial velocity and \(v\) is the final velocity.
- Displacement (s): Position relative to start (Vector).
- Velocity (u or v): How fast displacement changes (Vector).
- Acceleration (a): How fast velocity changes (Vector).
- Distance/Speed: The scalar versions (magnitude only).
Section 2: Visualizing Motion – Graphs
Graphs are crucial in M1. They allow us to visualize motion and calculate quantities without complex formulas.
2.1 Displacement-Time Graphs (\(s-t\))
This graph plots displacement (s) on the vertical axis against time (t) on the horizontal axis.
- What the Gradient Tells Us: The gradient (slope) of the \(s-t\) graph represents the Velocity. \[\text{Velocity} = \frac{\text{Change in } s}{\text{Change in } t}\]
- Key Shapes:
- A straight, diagonal line means Constant Velocity (constant gradient).
- A horizontal line means the particle is Stationary (velocity is zero).
- A curve means the velocity is changing, i.e., the particle is Accelerating.
2.2 Velocity-Time Graphs (\(v-t\))
This graph plots velocity (v) on the vertical axis against time (t) on the horizontal axis. This is the most important graph!
- What the Gradient Tells Us: The gradient of the \(v-t\) graph represents the Acceleration.
\[\text{Acceleration} = \frac{\text{Change in } v}{\text{Change in } t}\]
If the line is straight, the acceleration is constant (this is the only type of motion we study in depth in M1).
- What the Area Tells Us: The area under the \(v-t\) graph represents the Displacement.
If the area is below the t-axis (negative velocity), the displacement is negative, meaning the particle is moving backward relative to the chosen positive direction.
If you are asked to find the total distance traveled from a \(v-t\) graph, you must calculate the area of any parts below the axis separately and then add the magnitudes (positive values) together!
Section 3: The Toolkit – Constant Acceleration Equations (SUVAT)
When a particle moves with constant acceleration, we can use a set of five powerful formulas. These are often known by the variables they contain: SUVAT.
3.1 Defining the SUVAT Variables
Before using the equations, you must define what each letter stands for:
- s = Displacement (m)
- u = Initial Velocity (\(\text{ms}^{-1}\))
- v = Final Velocity (\(\text{ms}^{-1}\))
- a = Constant Acceleration (\(\text{ms}^{-2}\))
- t = Time (s)
Crucial Rule: These equations only work if acceleration (\(a\)) is constant.
3.2 The Five SUVAT Equations
Each equation is designed to solve a problem where one of the five variables is unknown or irrelevant (i.e., it is missing one of the five letters).
1. Formula missing s (Displacement): \[v = u + at\]
2. Formula missing a (Acceleration): \[s = \frac{(u+v)}{2} t\] (This is useful because \(\frac{(u+v)}{2}\) represents the average velocity.)
3. Formula missing v (Final Velocity): \[s = ut + \frac{1}{2} a t^2\]
4. Formula missing t (Time): \[v^2 = u^2 + 2as\] (Often used when time is not known or not needed.)
5. Formula missing u (Initial Velocity): \[s = vt - \frac{1}{2} a t^2\]
- Direction: Choose and clearly state which direction is positive (e.g., Up, Down, Right).
- List: Write down the five letters (s, u, v, a, t) and fill in the known values, being careful with positive (+) and negative (-) signs based on your chosen direction.
- Identify: Circle the variable you need to find, and cross out the variable you don't know and don't care about.
- Choose: Select the SUVAT equation that uses the variables you have and the variable you need to find.
- Solve: Substitute the values and calculate the answer.
Section 4: Key Applications – Vertical Motion and Tricky Scenarios
The most common application of SUVAT is movement under gravity (throwing an object up or dropping it down).
4.1 Motion Under Gravity
When an object is moving freely vertically (up or down), the only force acting on it (ignoring air resistance, which we usually do in M1) is gravity. This results in a constant acceleration.
- The acceleration due to gravity is denoted by g.
- In Edexcel exams, use \(g = 9.8 \text{ ms}^{-2}\) unless otherwise stated.
- Gravity always acts downwards.
Handling the Sign of a:
- Scenario A: If you choose UP to be the positive direction, then \(a = -9.8 \text{ ms}^{-2}\).
- Scenario B: If you choose DOWN to be the positive direction, then \(a = +9.8 \text{ ms}^{-2}\).
Always be consistent! If you throw a ball upwards with \(u = +10\), and you choose UP as positive, then \(a\) must be \(-9.8\).
4.2 The Peak Height and Changing Direction
When an object is thrown upwards and reaches its maximum height, it momentarily stops before falling back down.
Key fact to remember:
- At the maximum height, the Final Velocity (v) is zero (\(v=0\)).
4.3 Common Mistakes to Avoid
1. Confusing Displacement and Distance: If a particle goes 5m right and 5m left, the distance is 10m, but the displacement is 0m.
2. Forgetting Vector Signs: If you define right as positive, and acceleration is to the left (deceleration), you must input \(a\) as a negative number into the SUVAT equations.
3. Overlooking u=0 or v=0:
- "Starts from rest" means \(u = 0\).
- "Comes to rest" means \(v = 0\).
- "Maximum height/stops momentarily" means \(v = 0\) at that moment.
4. Incorrectly Using Time: If a question asks for the time taken to travel *down* after reaching maximum height, do not use the time taken for the whole journey. Solve the problem in two parts (Up, then Down).
Chapter Summary: Key Takeaways
Kinematics is all about describing motion mathematically. Your success depends on correctly identifying vectors and scalars, consistently choosing a positive direction, and mastering the use of the SUVAT equations.
- Scalars (Distance, Speed) have magnitude only.
- Vectors (Displacement, Velocity, Acceleration) have magnitude and direction (+/-).
- On a v-t graph: Gradient is Acceleration, Area is Displacement.
- SUVAT equations only work for Constant Acceleration.
- For vertical motion, always use \(a = \pm 9.8 \text{ ms}^{-2}\).
Keep practicing your sign conventions, and you will find these problems become second nature! Good luck!