Constant acceleration formulae

Welcome to Constant Acceleration!

In this chapter, we are looking at Kinematics in 1 Dimension. Kinematics is just a fancy word for "the study of motion." We’re going to learn how to predict where an object will be, how fast it’s going, and how long it takes to get there—provided its acceleration stays the same throughout the journey.

Think of it like this: If you are in a car and you press the gas pedal down and hold it in exactly the same position, your speed increases at a steady rate. This steady increase is what we call constant acceleration. Don't worry if this seems a bit abstract at first; once you learn the "SUVAT" system, it becomes much like solving a puzzle!

1. The Five Key Variables: Meet "SUVAT"

To solve these problems, we use five variables. We often call these the SUVAT variables to help us remember them. Each letter represents a specific physical quantity:

s = Displacement (The straight-line distance from the start point, measured in metres, \( \text{m} \)).
u = Initial Velocity (The speed at the very start of the timing, measured in \( \text{ms}^{-1} \)).
v = Final Velocity (The speed at the very end of the timing, measured in \( \text{ms}^{-1} \)).
a = Constant Acceleration (How much the velocity changes every second, measured in \( \text{ms}^{-2} \)).
t = Time (How long the movement takes, measured in seconds, \( \text{s} \)).

Quick Review: Direction Matters!

In 1D kinematics, objects can go forward or backward (or up and down). We usually pick one direction to be positive (like "Up" or "Right"). If an object moves in the opposite direction, its displacement and velocity become negative. If it’s slowing down in the positive direction, its acceleration is negative!

Key Takeaway: Before starting any problem, list out s, u, v, a, t and fill in the numbers you know. You usually need three pieces of information to find the other two.

2. The Constant Acceleration Formulae

These are the core tools of this chapter. You need to be able to use these fluently and understand where they come from.

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 choose the right formula:

Each formula is missing one of the SUVAT variables. To pick the right one, look at what you don't have and don't need:
- No s? Use formula 1.
- No a? Use formula 2.
- No v? Use formula 3.
- No u? Use formula 4.
- No t? Use formula 5.

Did you know? These formulae only work if acceleration is constant. if the acceleration changes (like a car shifting gears), you have to split the problem into separate parts or use calculus!

3. Deriving the Formulae

The MEI syllabus requires you to know how to derive these. The easiest way is using a Velocity-Time graph.

Imagine a graph where the velocity starts at \( u \) and reaches \( v \) after time \( t \).
Deriving \( v = u + at \): Since acceleration is the gradient of a velocity-time graph:
\( a = \frac{\text{change in velocity}}{\text{time}} = \frac{v - u}{t} \). Rearranging this gives \( at = v - u \), or \( v = u + at \).

Deriving \( s = \frac{1}{2}(u + v)t \): Displacement is the area under the graph. For a trapezoid (which is what the graph looks like), the area is the average of the parallel sides times the width:
\( \text{Area} = \frac{(u + v)}{2} \times t \). Hence, \( s = \frac{1}{2}(u + v)t \).

Key Takeaway: If you ever forget a formula in the exam, try sketching a quick Velocity-Time graph and calculating the gradient or area!

4. Working with Gravity (Free Fall)

One of the most common real-world examples of constant acceleration is an object falling under gravity. On Earth, we model this as a constant acceleration downward.

The Value of g: In your MEI exams, always use \( g = 9.8 \, \text{ms}^{-2} \) unless the question says otherwise.
Common Mistake: Students often forget that if they choose "Up" as the positive direction, then \( a = -9.8 \). If you drop an object from a cliff, it is often easier to choose "Down" as positive, so \( a = 9.8 \).

Step-by-Step for Gravity Problems:

1. Choose a direction to be positive (usually 'up' is best if the object is thrown up).
2. List your SUVAT: If it's dropped, \( u = 0 \). If it hits the ground, it doesn't mean \( v = 0 \) (it means the velocity a split second before impact).
3. Set \( a \): If 'up' is positive, \( a = -9.8 \).
4. Identify what you need and pick your formula.

Key Takeaway: At the maximum height of a thrown object, the vertical velocity (\( v \)) is always zero for a split second!

5. Problem Solving Tips & Common Pitfalls

Mechanics can be tricky because of the setup, not the math. Here is how to avoid the most common traps:

- Consistency: Ensure all your units match. If speed is in km/h and time is in seconds, you must convert km/h to m/s first.
- The "Hidden" Zeroes: Look for words like "starts from rest" (\( u = 0 \)) or "comes to a stop" (\( v = 0 \)).
- Deceleration: If an object is slowing down, its acceleration must have the opposite sign to its velocity.
- Two-part journeys: If a car accelerates then moves at a constant speed, you must do two separate SUVAT calculations. The "final velocity" of the first part becomes the "initial velocity" of the second part.

Summary of Key Points

• SUVAT stands for s (displacement), u (initial velocity), v (final velocity), a (acceleration), and t (time).
• These formulae only apply when acceleration is constant.
• Gravity on Earth is modeled as \( a = 9.8 \, \text{ms}^{-2} \) acting downwards.
• Always define your positive direction at the very start of your working.