Kinematics of motion in a straight line

Welcome to the World of Motion!

Hi there! Welcome to one of the most exciting parts of Mechanics: Kinematics. In this chapter, we are going to explore how objects move along a straight line. We aren't worried about why they are moving (we’ll save forces for later); we just want to describe the movement itself. Whether it’s a car braking at a red light or a sprinter zooming down a 100m track, the rules you’ll learn here apply to them all!

Don't worry if this seems tricky at first. Mechanics is often about visualizing the situation. Once you can "see" the motion in your head, the math becomes much easier to follow.

1. Scalars vs. Vectors: Why Direction Matters

Before we start calculating, we need to know the difference between two types of measurements. In Mechanics, just saying "how much" isn't always enough; often, we need to know "which way."

Scalars (The "How Much" quantities)

These only have magnitude (size). They don't care about direction.
1. Distance: How much ground you covered (e.g., "I walked 5 kilometers").
2. Speed: How fast you are going (e.g., "The car is doing 60 km/h").

Vectors (The "Which Way" quantities)

These have both magnitude AND direction.
1. Displacement (\(s\)): Your distance from a fixed starting point in a specific direction.
2. Velocity (\(v\)): Your speed in a specific direction (e.g., "20 m/s to the right").
3. Acceleration (\(a\)): The rate at which your velocity changes.

The Running Track Analogy:
If you run exactly one lap around a 400m circular track and end up back where you started:
• Your Distance is 400m.
• Your Displacement is 0m (because you are back at the start!).

Key Takeaway: In exam questions, always check if a value is "negative." A negative velocity usually just means the object is moving backward or in the opposite direction to what you chose as "positive."

2. Graphs of Motion

Graphs are a great way to "see" motion. There are two main types you need to master for the 9709 syllabus.

Displacement-Time (\(s-t\)) Graphs

• The Gradient (slope) of the line represents the Velocity.
• A straight diagonal line means constant velocity.
• A horizontal (flat) line means the object is stationary (not moving).
• A curved line means the velocity is changing (the object is accelerating).

Velocity-Time (\(v-t\)) Graphs

This is the most common graph in Paper 4! Memorize these two rules:
1. The Gradient represents the Acceleration.
2. The Area under the graph represents the Displacement.

Quick Tip: If the line on a \(v-t\) graph is sloping downwards, the object is slowing down. We call this deceleration or negative acceleration.

Key Takeaway: Always look at the axes first! A flat line on an \(s-t\) graph means "stopped," but a flat line on a \(v-t\) graph means "moving at a steady speed."

3. Constant Acceleration (The SUVAT Equations)

When an object moves with constant (unchanging) acceleration, we can use five special formulas. We call them the SUVAT equations because of the variables involved:

• \(s\) = Displacement
• \(u\) = Initial velocity (starting speed)
• \(v\) = Final velocity (ending speed)
• \(a\) = Constant acceleration
• \(t\) = Time taken

The Big Four Formulas:

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

How to solve SUVAT problems (Step-by-Step):

1. List your variables: Write down \(s, u, v, a, t\) and fill in what you know from the question.
2. Identify the "Target": What are you trying to find?
3. Choose the formula: Find the equation that uses your "knowns" and your "target," but not the variable you don't care about.
4. Check your signs: If an object is thrown upwards, and you choose "up" as positive, then acceleration due to gravity (\(g\)) must be negative (\(-10 m/s^2\)).

Did you know? In Cambridge 9709 Mechanics, we almost always use \(g = 10 m/s^2\) for acceleration due to gravity, unless the question says otherwise!

Key Takeaway: You can only use SUVAT if the acceleration is constant. If the acceleration changes, you must use Calculus!

4. Variable Acceleration (Using Calculus)

Sometimes acceleration isn't a nice, steady number. It might be a function of time, like \(a = 3t - 1\). When you see "t" in an expression for velocity or acceleration, it's time to use Calculus.

Moving "Down" (Differentiation)

If you have displacement and want velocity, or have velocity and want acceleration, you differentiate with respect to time (\(t\)).
• \(v = \frac{ds}{dt}\)
• \(a = \frac{dv}{dt}\)

Moving "Up" (Integration)

If you have acceleration and want velocity, or have velocity and want displacement, you integrate with respect to time (\(t\)).
• \(v = \int a \, dt\)
• \(s = \int v \, dt\)

Common Mistake: The constant of integration!
When you integrate, don't forget to add \(+ c\). You usually find the value of \(c\) by looking for "initial conditions" in the question (e.g., "at time \(t=0\), the velocity is \(5 m/s\)").

Quick Review Box:
• Displacement \(\rightarrow\) Velocity \(\rightarrow\) Acceleration (Differentiate!)
• Acceleration \(\rightarrow\) Velocity \(\rightarrow\) Displacement (Integrate!)

Key Takeaway: Calculus is the "master key" of kinematics. It works for all types of motion, whereas SUVAT only works for constant acceleration.

Final Summary for Revision

• Vectors have direction; Scalars do not.
• \(v-t\) Graphs: Gradient = Acceleration, Area = Displacement.
• SUVAT: Use only for constant acceleration. Pick your equation based on what is missing.
• Calculus: Use for variable acceleration. Remember to integrate to go from acceleration back to displacement, and always find your \(+ c\)!

You've got this! Practice drawing the graphs and listing your SUVAT variables for every problem, and you'll see your confidence grow.