Welcome to Kinematics!
Welcome to one of the most exciting parts of your A Level Mechanics course. Kinematics is the study of motion. Essentially, we are looking at how things move—whether it's a car braking, a ball being kicked, or a rocket launching. We don't worry about the forces causing the motion yet (that’s the next chapter!); we just focus on the path, the speed, and the timing.
Don’t worry if this seems a bit "physics-heavy" at first. We will break everything down into simple steps, and you’ll soon see that it’s all about a few key formulas and some clever logic.
1. The Language of Motion
Before we start calculating, we need to speak the same language. In kinematics, words that sound similar in everyday life have very specific meanings.
Scalars vs. Vectors
In A Level Maths, we distinguish between Scalars (which only have a size) and Vectors (which have a size AND a direction).
- Distance (Scalar): How far you have travelled in total. (e.g., 50 miles)
- Displacement, \(s\) (Vector): How far you are from your starting point in a straight line, including the direction. (e.g., 50 miles North)
- Speed (Scalar): How fast you are going. (e.g., 30 m/s)
- Velocity, \(v\) or \(u\) (Vector): Speed in a given direction. (e.g., +30 m/s or -30 m/s)
- Acceleration, \(a\) (Vector): The rate at which velocity changes.
Quick Review: Remember that Distance and Speed must always be positive. However, Displacement, Velocity, and Acceleration can be negative. A negative velocity just means you are moving in the opposite direction to what you chose as "positive" (usually left or down)!
Did you know? If you run exactly one lap around a 400m track and end up back where you started, your distance travelled is 400m, but your displacement is 0!
Key Takeaway: Always define which direction is positive (e.g., "Upwards is positive") before you start a problem.
2. Motion Graphs
Sometimes, a picture is worth a thousand equations. We use two main types of graphs to describe motion in a straight line.
Displacement-Time (\(s-t\)) Graphs
- The Gradient: The slope of the line represents the Velocity.
- A straight diagonal line means constant velocity.
- A horizontal line means the object is stationary (velocity = 0).
- A curve means the object is accelerating or decelerating.
Velocity-Time (\(v-t\)) Graphs
- The Gradient: The slope of the line represents the Acceleration.
- The Area Under the Graph: This represents the Displacement (or distance travelled).
Memory Aid: Think of the word G-A-V.
Gradient of Acceleration-time (not used here) isn't helpful, but...
Gradient of Velocity-time = Acceleration.
Area of Velocity-time = Displacement.
Common Mistake: Students often forget that area "below" the x-axis on a \(v-t\) graph counts as negative displacement! If you want total distance, treat that area as positive. If you want displacement, keep it negative.
3. Constant Acceleration (SUVAT)
When an object moves in a straight line with constant (unchanging) acceleration, we can use the famous SUVAT equations. These letters stand for:
\(s\) = Displacement
\(u\) = Initial Velocity
\(v\) = Final Velocity
\(a\) = Constant Acceleration
\(t\) = Time
The SUVAT 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. \(v^2 = u^2 + 2as\)
How to solve SUVAT problems:
- List your variables: Write down \(s, u, v, a, t\) and fill in what you know. You need 3 pieces of information to find the other 2.
- Check units: Ensure everything is in meters (m) and seconds (s).
- Pick the formula: Choose the equation that uses the variables you have and the one you want to find.
- Solve: Plug in the numbers and rearrange.
Example: A car starts from rest (\(u=0\)) and accelerates at \(2 \text{ m/s}^2\) for 5 seconds. How far does it go?
We have \(u=0, a=2, t=5\). We want \(s\). We use \(s = ut + \frac{1}{2}at^2\).
\(s = (0)(5) + \frac{1}{2}(2)(5^2) = 25\text{m}\).
Quick Tip: If a question says an object starts "from rest," it means \(u = 0\). If it "comes to a stop," it means \(v = 0\).
4. Working with Vectors in 2D
Sometimes objects don't just move in a line; they move across a plane. We use i and j notation (or column vectors) to describe this.
The SUVAT equations still work! You just apply them to the vector components. For example:
\(\mathbf{v} = \mathbf{u} + \mathbf{a}t\)
\(\mathbf{r} = \mathbf{u}t + \frac{1}{2}\mathbf{a}t^2\)
Here, \(\mathbf{r}\) is the position vector. If an object starts at a position \(\mathbf{r_0}\), its position at time \(t\) is:
\(\mathbf{r} = \mathbf{r_0} + \mathbf{u}t + \frac{1}{2}\mathbf{a}t^2\)
5. Kinematics with Calculus
What if the acceleration is not constant? SUVAT won't work! Instead, we use Calculus (Differentiation and Integration).
The Calculus Ladder:
To move DOWN the ladder (finding rates of change), we Differentiate with respect to time (\(t\)):
Displacement (\(s\) or \(r\))
\(\downarrow\) Differentiate
Velocity (\(v\)) \( = \frac{ds}{dt}\)
\(\downarrow\) Differentiate
Acceleration (\(a\)) \( = \frac{dv}{dt} = \frac{d^2s}{dt^2}\)
To move UP the ladder, we Integrate with respect to time (\(t\)):
Acceleration (\(a\))
\(\downarrow\) Integrate
Velocity (\(v\)) \( = \int a \, dt\)
\(\downarrow\) Integrate
Displacement (\(s\)) \( = \int v \, dt\)
Important! Whenever you integrate, don't forget the constant of integration (\(+c\)). You usually find this by looking for "initial conditions" in the question (like "at \(t=0, v=2\)").
Don't worry if this seems tricky: Just remember: "Differentiate to go from displacement to acceleration; Integrate to go back up."
6. Projectiles
A projectile is an object moving through the air under only the force of gravity. We model this by splitting the motion into two independent parts: Horizontal and Vertical.
Horizontal Motion (The easy part):
- There is no acceleration (\(a = 0\)).
- Velocity remains constant throughout the flight.
- Formula: \(x = u_x \times t\) (Distance = Speed \(\times\) Time).
Vertical Motion (The SUVAT part):
- Acceleration is constant due to gravity: \(a = -9.8 \text{ m/s}^2\) (if upwards is positive).
- Use SUVAT equations for this direction.
Key Facts for Projectiles:
- Time is the Bridge: The time \(t\) is the same for both horizontal and vertical components. Usually, you find \(t\) using one direction and use it in the other.
- At the Peak: At the highest point of the path, the vertical velocity is zero (\(v_y = 0\)).
- Symmetry: If a ball is kicked from the ground and lands on the ground, the time to reach the peak is exactly half the total flight time.
Quick Review: If an object is launched at an angle \(\theta\) with speed \(U\):
Horizontal initial velocity: \(u_x = U \cos\theta\)
Vertical initial velocity: \(u_y = U \sin\theta\)
Key Takeaway: Never mix horizontal and vertical values in the same SUVAT equation! Keep them in separate columns on your page.
Common Pitfalls to Avoid
- Mixing Units: Always convert km/h to m/s. (Multiply by 1000, divide by 3600).
- Signs: If you decide "Up" is positive, then gravity (\(g\)) must be \(-9.8\). If an object is falling, its displacement \(s\) will be negative.
- SUVAT for variable acceleration: If you see an equation for velocity like \(v = 3t^2 + 2\), do not use SUVAT. You must use calculus!
- Calculator Mode: When resolving angles for projectiles (\(\sin\theta, \cos\theta\)), make sure your calculator is in Degrees (unless the question uses Radians).
You've got this! Kinematics is just a giant puzzle. Once you identify your "knowns" and your "unknowns," the formulas do the heavy lifting for you.