Welcome to Constant Acceleration!
In this chapter, we are going to look at how objects move when they are speeding up or slowing down at a steady rate. Whether it’s a car pulling away from a green light or a ball dropped from a window, these "SUVAT" formulae are your secret weapons for predicting exactly where an object will be and how fast it will be going.
Don't worry if the algebra looks a bit crowded at first. Once you learn the "SUVAT" trick, you'll see that these problems are just like solving a mini-mystery where you have some clues and need to find the missing piece!
1. Meet the SUVAT Variables
Before we use the equations, we need to know the "players" involved. We call these the SUVAT variables because of the letters we use to represent them. Each one describes a different part of the motion in one dimension (a straight line).
s = Displacement: This is the straight-line distance from the starting point (measured in metres, \(m\)).
u = Initial Velocity: How fast the object was going at the very start (measured in \(m s^{-1}\)).
v = Final Velocity: How fast the object is going at the end of the time period (measured in \(m s^{-1}\)).
a = Acceleration: The steady rate at which the velocity is changing (measured in \(m s^{-2}\)).
t = Time: How long the motion lasted (measured in seconds, \(s\)).
Quick Tip: The Displacement Trap
Remember that displacement (s) is not always the same as distance. If you throw a ball up and catch it in the same place, its total distance might be 10 metres, but its displacement is 0 because it ended up exactly where it started!
Key Takeaway: To solve a constant acceleration problem, you need to identify as many of these five variables as possible from the question.
2. When can we use these formulae?
This is the most important rule in this chapter: You can ONLY use these formulae if the acceleration is CONSTANT.
If a car is accelerating at \(2 m s^{-2}\) for the whole journey, SUVAT works perfectly. If the car is jerky—speeding up, then slowing down, then speeding up again—SUVAT will give you the wrong answer. In those cases, you would need calculus instead!
Did you know?
The most common example of constant acceleration is gravity. On Earth, if we ignore air resistance, everything falls with a constant acceleration of approximately \(g = 9.8 m s^{-2}\).
3. The "Big Five" Formulae
According to your MEI H630 syllabus, you need to be familiar with these five equations. Each one is missing exactly one of the SUVAT variables. This is helpful because if you don't know "s", you pick the equation without "s" in it!
1. \( v = u + at \) (Missing s)
2. \( s = \frac{1}{2}(u + v)t \) (Missing a)
3. \( s = ut + \frac{1}{2}at^2 \) (Missing v)
4. \( s = vt - \frac{1}{2}at^2 \) (Missing u)
5. \( v^2 - u^2 = 2as \) (Missing t)
Key Takeaway: You don't need to memorize which one is "missing" what; just look at the variables you have and the one you want to find, then pick the equation that fits!
4. How to Derive the Formulae
The syllabus requires you to know how to derive these. Don't let the word "derive" scare you—it just means "show where they came from." It's actually quite logical!
Deriving \( v = u + at \)
This comes straight from the definition of acceleration. Acceleration is the change in velocity divided by time:
\( a = \frac{v - u}{t} \)
Multiply both sides by \(t\):
\( at = v - u \)
Add \(u\) to both sides:
\( v = u + at \)
Deriving \( s = \frac{1}{2}(u + v)t \)
For constant acceleration, the average velocity is exactly halfway between the start and end velocities:
Average Velocity = \( \frac{u + v}{2} \)
Since Displacement = Average Velocity \(\times\) Time:
\( s = (\frac{u + v}{2})t \)
Key Takeaway: All the other formulae are found by substituting these two into each other to eliminate the variable you don't want.
5. Solving Problems: Step-by-Step
If you find mechanics tricky, follow this checklist every single time. It works!
Step 1: Draw a diagram. Draw a box or a dot and an arrow to show which way is positive (usually right or up).
Step 2: List your SUVAT variables. Write "S, U, V, A, T" in a column and fill in what you know from the text.
Step 3: Identify the goal. Put a question mark next to the variable the question is asking for.
Step 4: Pick your equation. Choose the formula that uses your "knowns" and your "question mark."
Step 5: Solve. Plug in the numbers and calculate the answer.
Example: The Braking Car
A car travelling at \(20 m s^{-1}\) brakes steadily to a stop over a distance of 50m. Find the acceleration.
s = 50
u = 20
v = 0 (because it comes to a "stop")
a = ?
t = (not mentioned)
We use \( v^2 - u^2 = 2as \):
\( 0^2 - 20^2 = 2 \times a \times 50 \)
\( -400 = 100a \)
\( a = -4 m s^{-2} \)
The acceleration is negative because the car is slowing down!
6. Common Mistakes to Avoid
1. Mixing Directions: If you decide "Up" is positive, then gravity (\(a\)) must be -9.8 because it pulls "Down." If you mix these up, your object might accidentally fly into space in your calculations!
2. Units: Always check that your time is in seconds and distance is in metres. If the question says "km/h," convert it before you start.
3. Squared Terms: In the formula \( s = ut + \frac{1}{2}at^2 \), only the \(t\) is squared. A very common mistake is squaring the whole \((at)\) part.
4. Assuming u = 0: Only use \(u = 0\) if the question says "starts from rest" or "dropped."
Quick Review Box:
- Constant acceleration is the only time SUVAT works.
- Direction matters: Use signs (+/-) consistently.
- Deceleration is just negative acceleration.
- At rest means velocity is zero.
Summary: You now have the tools to describe motion in a straight line! By identifying your SUVAT variables and picking the right equation, you can solve almost any kinematics problem involving steady speed changes.