Welcome to the World of Motion: Kinematics

Welcome to Kinematics! This chapter is part of your Paper 1 studies for AQA AS Mathematics. In simple terms, kinematics is the study of how things move. We don’t worry about what started the movement (that’s for the "Forces" chapter); we just focus on describing the path, speed, and timing of an object's journey.

Think of yourself as a sports commentator describing a 100m sprint. You need to know where the runner starts, how fast they are going, and if they are speeding up at the end. That is exactly what we are going to learn here!

1. The Language of Kinematics

In mechanics, we use specific words to describe motion. It is very important to know the difference between scalars (which only have size) and vectors (which have size and direction).

Distance vs. Displacement

  • Distance Travelled (Scalar): This is the total ground you've covered. If you walk 5m forward and 5m back, your distance is 10m.
  • Displacement (Vector): This is how far you are from your starting point. In the example above, your displacement is 0m because you ended up exactly where you started!

Speed vs. Velocity

  • Speed (Scalar): How fast an object is moving.
  • Velocity (Vector): Speed in a given direction. If a car travels at 30 mph North, that is its velocity. If it turns around and goes 30 mph South, its speed is the same, but its velocity has changed to -30 mph (relative to North).

Acceleration

Acceleration is the rate at which velocity changes. If you are "speeding up," "slowing down," or "changing direction," you are accelerating. In this course, we often deal with constant acceleration, like an object falling under gravity.

Quick Review:
Displacement (\(s\)): Where are you relative to the start?
Velocity (\(v\)): How fast are you going and in what direction?
Acceleration (\(a\)): How quickly is your velocity changing?

2. Motion Graphs

Graphs are a great way to "see" motion. There are two main types you need to master.

Displacement-Time Graphs

This graph shows how far an object has moved from the start over time.

  • The Gradient (Slope) = Velocity.
  • A steep line means high velocity.
  • A flat horizontal line means the object is stationary (velocity is zero).
  • A curve means the velocity is changing (the object is accelerating).

Velocity-Time Graphs

This graph shows how the velocity changes over time.

  • The Gradient (Slope) = Acceleration.
  • The Area under the graph = Displacement (Distance travelled from the start).
  • A line below the x-axis means the object is moving in the opposite direction.

Don’t worry if this seems tricky at first! Just remember: to find the "next" thing in the sequence (Displacement \(\rightarrow\) Velocity \(\rightarrow\) Acceleration), you look at the gradient. To go backwards, you look at the area.

Key Takeaway: On a velocity-time graph, "Area = Distance" and "Slope = Acceleration".

3. Constant Acceleration Formulae (SUVAT)

When an object moves in a straight line with constant acceleration, we can use five special equations. We call them the SUVAT equations because of the variables involved:

  • \(s\) = Displacement (m)
  • \(u\) = Initial velocity (m/s) — Think: "u" comes before "v" in the alphabet, so it's the start speed.
  • \(v\) = Final velocity (m/s)
  • \(a\) = Acceleration (m/s\(^2\))
  • \(t\) = Time (s)

The Five Equations:

1. \(v = u + at\)

2. \(s = ut + \frac{1}{2}at^2\)

3. \(s = vt - \frac{1}{2}at^2\)

4. \(v^2 = u^2 + 2as\)

5. \(s = \frac{1}{2}(u + v)t\)

How to solve SUVAT problems:

  1. List your variables: Write down \(s, u, v, a, t\) and fill in what you know from the question.
  2. Identify what you need: Mark the variable you are trying to find.
  3. Choose the equation: Find the equation that uses the three things you know and the one thing you want to find.

Common Mistake to Avoid: Always check your directions! If you decide that "Up" is positive, then "Down" (like gravity) must be negative. Gravity on Earth is usually taken as \(a = -9.8 \text{ m/s}^2\) if up is positive.

4. Kinematics and Calculus

What if the acceleration is not constant? This is where calculus comes in! We use differentiation to find rates of change and integration to find total change.

Going "Down" (Differentiation)

If you have a formula for displacement (\(r\) or \(s\)), you can differentiate it to find the others:

  • Velocity: \(v = \frac{dr}{dt}\)
  • Acceleration: \(a = \frac{dv}{dt} = \frac{d^2r}{dt^2}\)

Going "Up" (Integration)

If you have a formula for acceleration, you can integrate it to find the others:

  • Velocity: \(v = \int a \, dt\)
  • Displacement: \(r = \int v \, dt\)

Did you know? When you integrate, don't forget the constant of integration (+C)! In mechanics, this \(+C\) is often the initial velocity (\(u\)) or the initial position.

Step-by-Step Example:

If \(v = 3t^2 + 2\), find the displacement after 2 seconds (assuming it starts at 0).
1. Integrate the velocity: \(s = \int (3t^2 + 2) \, dt\)
2. This gives: \(s = t^3 + 2t + C\)
3. Since it starts at 0 (\(s=0\) when \(t=0\)), \(C = 0\).
4. Plug in \(t = 2\): \(s = (2)^3 + 2(2) = 8 + 4 = 12\text{m}\).

Summary Takeaway:
S \(\rightarrow\) V \(\rightarrow\) A: Differentiate (find the gradient).
A \(\rightarrow\) V \(\rightarrow\) S: Integrate (find the area).

Final Tips for Success

  • Units Matter: Always ensure your units are consistent (usually meters, seconds, and kg).
  • Draw a Diagram: Even a simple line representing the path of the object can help you visualize which way is positive and which is negative.
  • Read carefully: If a question says "starts from rest," it means \(u = 0\). If it says "comes to a stop," it means \(v = 0\).