Kinematics and Equations of Motion

TL;DR

Kinematics describes motion without considering its causes. It focuses on displacement, velocity, and acceleration. We use a set of five "suvat" equations to solve problems involving constant acceleration.

1. The Mental Model

Imagine you're tracking a car. Kinematics is just about describing how it moves – its starting point, how fast it goes, and if it speeds up or slows down. We're not worried about why it moves, like the engine or friction.

2. The Core Material

Kinematics is all about describing motion. The key quantities we track are:

• Displacement (s): This is the straight-line distance and direction from a starting point to an ending point. It's a vector quantity. Don't confuse it with total distance travelled, which is just the path length.
• Initial Velocity (u): How fast something is moving and in what direction at the start of the motion we're interested in. Also a vector.
• Final Velocity (v): How fast something is moving and in what direction at the end of the motion we're interested in. Also a vector.
• Acceleration (a): The rate at which velocity changes. If velocity is constant, acceleration is zero. If something speeds up or slows down, it's accelerating. Also a vector.
• Time (t): The duration over which the motion occurs. This is a scalar quantity (it only has magnitude, no direction).

The "suvat" Equations

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When acceleration is constant, we can use a set of five equations, often called "suvat" equations (after the letters representing the variables). You'll typically be given three variables and asked to find a fourth.

Here's a breakdown of the relationships:

graph TD
    Start["Given 3 variables"] --> WhichEq["Which equation to use?"]

    subgraph Equations
        Eq1["v = u + at"]
        Eq2["s = ut + ½at²"]
        Eq3["s = vt - ½at²"]
        Eq4["v² = u² + 2as"]
        Eq5["s = ½(u + v)t"]
    end

    WhichEq --> HasNoS{No 's' needed?}
    HasNoS -- Yes --> Eq1

    WhichEq --> HasNoV{No 'v' needed?}
    HasNoV -- Yes --> Eq2

    WhichEq --> HasNoU{No 'u' needed?}
    HasNoU -- Yes --> Eq3

    WhichEq --> HasNoT{No 't' needed?}
    HasNoT -- Yes --> Eq4

    WhichEq --> HasNoA{No 'a' needed?}
    HasNoA -- Yes --> Eq5

    Eq1 --> Solution
    Eq2 --> Solution
    Eq3 --> Solution
    Eq4 --> Solution
    Eq5 --> Solution

    Solution["Solve for unknown variable"]

Remember:
* Direction matters: If motion is in one direction, make it positive. Motion in the opposite direction is negative. Acceleration can also be positive (speeding up in the positive direction) or negative (slowing down in the positive direction, or speeding up in the negative direction).
* Units: Always use consistent units (e.g., meters for displacement, meters per second for velocity, meters per second squared for acceleration, and seconds for time).

3. Worked Example

A car accelerates uniformly from rest to 20 m/s in 8 seconds. How far did it travel during this time?

1. List knowns and unknowns:

   • Initial velocity, u = 0 m/s (since it starts from rest)
   • Final velocity, v = 20 m/s
   • Time, t = 8 s
   • Displacement, s = ?
   • Acceleration, a = ? (we don't need it here)

2. Choose the right equation: We need 's', and we know 'u', 'v', and 't'. Looking at the diagram or equations, s = ½(u + v)t doesn't involve 'a', which is perfect.

3. Substitute and solve:

   • s = ½(0 + 20) * 8
   • s = ½(20) * 8
   • s = 10 * 8
   • s = 80 m

The car travelled 80 meters.

4. Key Takeaways

• Kinematics describes motion using displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t).
• The "suvat" equations (v = u + at, s = ut + ½at², s = vt - ½at², v² = u² + 2as, s = ½(u + v)t) are only valid for constant acceleration.
• Direction is crucial; assign positive and negative signs consistently for vector quantities.
• Always ensure all quantities are in consistent units before performing calculations.
• You typically need at least three known variables to solve for an unknown using the "suvat" equations.

Common Mistakes to Avoid:

Photo by KATRIN BOLOVTSOVA on Pexels

• Confusing distance with displacement: Displacement is a straight line from start to finish, distance is the total path length.
• Forgetting about direction: Not assigning negative signs for motion or acceleration in the opposite direction.
• Using a = 0 when it's not: Don't assume acceleration is zero unless explicitly stated (e.g., "constant velocity").
• Mixing units: Using km/h for velocity with seconds for time will lead to incorrect answers. Convert everything to base units (meters, seconds) first.

5. Now Try It

A ball is thrown vertically upwards with an initial velocity of 15 m/s. Assuming air resistance is negligible and the acceleration due to gravity is -9.8 m/s² (negative because it acts downwards), how long does it take for the ball to reach its highest point?

What to do:
1. List all known variables and what you need to find.
2. Think about what happens to the ball's velocity at its highest point.
3. Select the appropriate "suvat" equation.
4. Solve for the unknown time.

Success looks like: You should find that the ball takes approximately 1.53 seconds to reach its highest point.