Kinematics

Welcome to the World of Kinematics!

Hello! Today, we are diving into Kinematics. This is the branch of mechanics that describes how objects move. Think of it as the "storytelling" of motion—we describe the speed, direction, and path of an object without worrying about the forces (like pushes or pulls) that caused the movement. Whether it's a car braking at a traffic light or a footballer kicking a ball, kinematics helps us predict exactly where it will be and when.

Don't worry if mechanics feels a bit "physics-heavy" at first. We will break everything down into simple steps and clear rules!

1. The Language of Motion

Before we start calculating, we need to speak the language. In Kinematics, we distinguish between Scalars (just a size) and Vectors (size AND direction).

Distance vs. Displacement

• Distance (Scalar): How much ground an object has covered. (e.g., "I walked 5 miles").
• Displacement, \(s\) (Vector): How far out of place an object is; it is the object's overall change in position. (e.g., "I am 3 miles North of my house").

Speed vs. Velocity

• Speed (Scalar): How fast an object is moving.
• Velocity, \(v\) or \(u\) (Vector): Speed in a given direction.

Acceleration

Acceleration, \(a\) (Vector), is the rate at which velocity changes. If you speed up, slow down (decelerate), or change direction, you are accelerating!

Quick Review Box:
Position: Where you are.
Displacement: Where you are relative to the start.
Velocity: Rate of change of displacement.
Acceleration: Rate of change of velocity.

Key Takeaway: Always check if a question asks for distance or displacement. If you run one lap of a 400m track, your distance is 400m, but your displacement is 0m because you are back where you started!

2. Seeing Motion: Kinematics Graphs

Sometimes a picture is worth a thousand equations. We primarily use two types of graphs:

Displacement-Time Graphs

• The gradient (slope) of the line represents the velocity.
• A flat horizontal line means the object is stationary.
• A straight diagonal line means constant velocity.

Velocity-Time Graphs

• The gradient represents the acceleration.
• The area under the graph represents the displacement.

Step-by-Step: Finding Distance from a V-T Graph
1. Identify the shapes under the line (usually triangles and rectangles).
2. Calculate the area of each shape.
3. Add them together to find the total displacement.

Analogy: Think of a V-T graph like a tap filling a bucket. The 'Velocity' is how fast the water flows, and the 'Area' (Displacement) is the total amount of water in the bucket.

3. Constant Acceleration: The SUVAT Equations

When an object moves in a straight line with constant acceleration, we use the famous SUVAT equations. Don't worry if these look intimidating; you'll get a formula sheet, but practicing them makes them feel like second nature!

The letters stand for:
s = displacement
u = initial velocity
v = final velocity
a = constant acceleration
t = time

The Big Five Equations:

1. \(v = u + at\)
2. \(s = \frac{1}{2}(u + v)t\)
3. \(s = ut + \frac{1}{2}at^2\)
4. \(s = vt - \frac{1}{2}at^2\)
5. \(v^2 = u^2 + 2as\)

Common Mistake to Avoid: These equations only work if acceleration is constant. If the acceleration changes (e.g., \(a = 3t\)), you must use calculus instead!

Key Takeaway: In any SUVAT problem, you usually know three pieces of information and need to find a fourth. Write down "S, U, V, A, T" on the side of your paper and fill in what you know first.

4. Calculus in Kinematics

What if acceleration isn't constant? This is where calculus saves the day! We use differentiation to go "down" the chain and integration to go "up".

The Chain of Motion:

Displacement \(s\) \(\rightarrow\) Velocity \(v\) \(\rightarrow\) Acceleration \(a\)

• To move right (find velocity from displacement, or acceleration from velocity), Differentiate with respect to time \(t\).
  \(v = \frac{ds}{dt}\) and \(a = \frac{dv}{dt} = \frac{d^2s}{dt^2}\)


• To move left (find velocity from acceleration, or displacement from velocity), Integrate with respect to time \(t\).
  \(v = \int a \, dt\) and \(s = \int v \, dt\)

Memory Aid: Remember "D-V-A".
Displacement \(\rightarrow\) Velocity \(\rightarrow\) Acceleration.
Going "Down" the list? Differentiate!
Going "Up" the list? Integrate!

Important Tip: When you integrate, don't forget the constant of integration (+C)! You usually find this by using the "initial conditions" (e.g., "at \(t = 0\), the object was at the origin").

5. Motion Under Gravity & Projectiles

When we throw something into the air, gravity pulls it down. In the AQA syllabus, we assume gravity \(g\) is \(9.8 \, ms^{-2}\) and acts vertically downwards.

Vertical Motion

If you drop an object, its initial velocity \(u = 0\) and its acceleration \(a = 9.8\). If you throw it up, its acceleration is \(a = -9.8\) (because it's slowing down).

Projectiles (2D Motion)

When an object is kicked at an angle, it moves both horizontally and vertically. The secret to solving these is to treat the horizontal and vertical components separately.

• Horizontal: There is no acceleration (\(a = 0\)). The velocity stays constant the whole time!
• Vertical: Use SUVAT with \(a = -9.8 \, ms^{-2}\).

Did you know? At the very highest point of a projectile's path, its vertical velocity is zero for a split second. This is a vital clue for many exam questions!

Step-by-Step: Projectile Problems
1. Resolve the initial velocity into components: \(u_x = u \cos(\theta)\) and \(u_y = u \sin(\theta)\).
2. List your SUVAT variables for the vertical direction.
3. Use the vertical motion to find the time \(t\).
4. Use that same time \(t\) in the horizontal direction (Distance = Speed \(\times\) Time) to find the range.

Key Takeaway: Time is the "bridge" between horizontal and vertical motion. It is the only variable that is the same for both.

Summary Quick-Check

• Are you using vectors (with direction) or scalars?
• Is acceleration constant? Use SUVAT.
• Is acceleration changing? Use Calculus (Differentiation/Integration).
• On a Velocity-Time graph, is the gradient acceleration and the area displacement?
• For projectiles, have you split the motion into horizontal (constant speed) and vertical (gravity)?

You've got this! Kinematics is all about being organized with your variables. Keep practicing those SUVAT substitutions!