Introduction to Kinematics

Welcome to the study of Kinematics! In the world of Mathematics: Analysis and Approaches, kinematics is the study of motion. We aren't worried about why something is moving (that's physics!); we are focused on how it moves using the powerful tools of Calculus. Whether it's a car driving along a straight road or a particle moving along an axis, we use displacement, velocity, and acceleration to describe its journey.

If you've already studied differentiation and integration, you've done the hard part. Kinematics is simply applying those rules to time-based functions. Let's dive in!

1. The Three Core Variables

In IB Mathematics, we usually look at motion in a straight line. We track three main quantities, all of which are functions of time \(t\):

  • Displacement \(s(t)\): This is the object's position relative to a fixed origin. Unlike distance, displacement can be negative. If \(s(t) > 0\), the object is to the right of the origin; if \(s(t) < 0\), it is to the left.
  • Velocity \(v(t)\): This is the rate of change of displacement. It tells us how fast the object is moving and in which direction.
    • If \(v(t) > 0\), the object is moving in the positive direction (usually right).
    • If \(v(t) < 0\), the object is moving in the negative direction (usually left).
    • If \(v(t) = 0\), the object is instantaneously at rest.
  • Acceleration \(a(t)\): This is the rate of change of velocity. It tells us how the velocity is changing over time.

Quick Note: We always assume \(t \geq 0\) because time doesn't go backward in these problems!

2. The Calculus Connection

The beauty of kinematics lies in how these three variables are linked through Differentiation and Integration. Think of them as a ladder:

Moving Down the Ladder (Differentiation)

If you have the displacement and want to find velocity, or have velocity and want to find acceleration, you differentiate with respect to time \(t\):

\(v(t) = \frac{ds}{dt} = s'(t)\)

\(a(t) = \frac{dv}{dt} = v'(t) = s''(t)\)

Moving Up the Ladder (Integration)

If you have acceleration and want velocity, or have velocity and want displacement, you integrate with respect to time \(t\):

\(v(t) = \int a(t) \, dt\)

\(s(t) = \int v(t) \, dt\)

Important: Whenever you integrate to find \(v(t)\) or \(s(t)\), don't forget the constant of integration \(+ C\)! You will usually be given "initial conditions" (like the position at \(t = 0\)) to help you solve for \(C\).

Key Takeaway: Differentiation takes you from displacement \(\to\) velocity \(\to\) acceleration. Integration takes you back the other way.

3. Displacement vs. Total Distance Travelled

This is a classic "trap" in IB exams. There is a huge difference between where you end up and how far you actually walked.

Displacement over a time interval \([t_1, t_2]\)

This is simply the change in position. It is calculated as:

\(\int_{t_1}^{t_2} v(t) \, dt = s(t_2) - s(t_1)\)

Total Distance Travelled over a time interval \([t_1, t_2]\)

This counts every step you took, even if you turned around and went back. Because distance cannot be negative, we use the absolute value of velocity:

\(\text{Total Distance} = \int_{t_1}^{t_2} |v(t)| \, dt\)

How to solve this without a calculator (Paper 1):
1. Find when the particle is at rest by solving \(v(t) = 0\).
2. Check if any of those times fall within your interval \([t_1, t_2]\).
3. Split your integral into sections where the velocity is positive and where it is negative, and make the negative results positive before adding them up.

How to solve this with a calculator (Paper 2):
Simply use the absolute value function on your GDC: \(\int_{t_1}^{t_2} \text{abs}(v(t)) \, dt\). Easy!

4. Speed vs. Velocity

While people use these terms interchangeably in real life, they are different in math:

  • Velocity is a vector (it has direction). It can be \(-5 \, ms^{-1}\).
  • Speed is the magnitude of velocity. It is always positive.
  • Formula: \(\text{Speed} = |v(t)|\).

Example: If a particle has a velocity of \(-10 \, ms^{-1}\), its speed is \(10 \, ms^{-1}\).

5. Common Phrases and What They Mean

IB exam questions often use specific wording. Here is a "translator" for you:

  • "Initially": This means when time \(t = 0\).
  • "At the origin": This means displacement \(s(t) = 0\).
  • "At rest": This means velocity \(v(t) = 0\).
  • "Changing direction": This happens when velocity \(v(t)\) changes sign (from positive to negative or vice versa). This usually occurs when \(v(t) = 0\).
  • "Constant velocity": This means acceleration \(a(t) = 0\).

6. Summary Table for Quick Review

To find... | What to do...
Velocity from Displacement | Differentiate: \(v = s'\)
Acceleration from Velocity | Differentiate: \(a = v'\)
Velocity from Acceleration | Integrate: \(v = \int a \, dt\)
Displacement from Velocity | Integrate: \(s = \int v \, dt\)
Total Distance | Integrate absolute velocity: \(\int |v| \, dt\)

Did you know?
The rate of change of acceleration is actually called "jerk." While it's not in your syllabus, it’s why your head snaps back when a car suddenly speeds up! For your IB exam, though, you only need to go as far as acceleration \(a(t)\).

Common Mistakes to Avoid

  • Forgetting \(+ C\): When finding \(v(t)\) from \(a(t)\), or \(s(t)\) from \(v(t)\), always include the constant and use the given information to find it.
  • Mixing up Distance and Displacement: Always check if the question asks for "displacement" or "total distance." If it's total distance, look for those turning points where \(v(t) = 0\).
  • Radians vs. Degrees: Kinematics problems often involve trigonometric functions (like \(v(t) = \sin(t)\)). Always ensure your calculator is in Radians mode for Calculus!

Don't worry if this seems tricky at first! Just remember the "ladder" of \(s \to v \to a\). Once you know which direction you are moving on that ladder, the calculus rules you already know will guide you to the answer.