Welcome to the World of Scalars and Vectors!
In your GCSEs, you probably treated most numbers just as... well, numbers. But in AS Level Physics, we need to be more precise. Imagine someone tells you that a "treasure is hidden 50 meters away." You’d be frustrated because you don’t know which direction to walk!
In this chapter, we will learn how to distinguish between quantities that only have a size and those that also have a direction. This is a foundational skill that you will use in every single module of Physics A, from mechanics to electricity.
1. Scalar and Vector Quantities
Every physical quantity we measure can be put into one of two "buckets": Scalars or Vectors.
What is a Scalar?
A scalar quantity has magnitude (size) only. It does not have a direction.
Examples: Mass (5 kg), Time (10 s), Temperature (20°C), Speed (30 m/s), and Distance (100 m).
What is a Vector?
A vector quantity has both magnitude (size) AND direction.
Examples: Force (10 N Down), Velocity (30 m/s North), Displacement (100 m East), and Acceleration (9.81 m/s\(^2\) towards the Earth).
Quick Review Box:
• Scalar = Size only.
• Vector = Size + Direction.
Memory Aid: Think of Scalar as Size and Vector as Velocity (which needs a direction!).
Did you know? Distance is a scalar (how much ground you covered), but Displacement is a vector (how far you are from where you started, in a straight line). If you run one lap of a 400m track, your distance is 400m, but your displacement is 0m because you ended up back where you started!
Key Takeaway: Always ask yourself: "Does the direction of this number matter?" If yes, it's a vector!
2. Vector Addition and Subtraction
Because vectors have direction, we can't always just add the numbers together like 1 + 1 = 2. We have to look at which way they are pointing.
Vectors in a Straight Line
If two vectors are in the same direction, simply add them.
Example: A boat moving at 5 m/s with a 2 m/s current behind it has a resultant velocity of \( 5 + 2 = 7 \) m/s.
If they are in opposite directions, subtract the smaller from the larger.
Example: If you pull a box with 10 N to the right, and your friend pulls with 3 N to the left, the resultant force is \( 10 - 3 = 7 \) N to the right.
Common Mistake to Avoid:
Don't forget to state the direction in your final answer! A vector answer is incomplete without it. Instead of just writing "7 N," write "7 N to the right."
3. The Vector Triangle (Resultants)
What happens if vectors aren't in a straight line? For example, one force pulls North and another pulls East. To find the resultant (the single vector that has the same effect as the others combined), we use a vector triangle.
The "Tip-to-Tail" Rule
Don't worry if this seems tricky at first! Just follow these steps:
1. Draw your first vector as an arrow.
2. Draw the second vector starting from the tip (the pointy end) of the first one.
3. The resultant is the arrow drawn from the tail of the first to the tip of the last.
Finding the Resultant: Two Methods
1. Calculation (for right-angled vectors):
If the two vectors are at 90° to each other, use Pythagoras' Theorem:
\( a^2 + b^2 = c^2 \)
To find the angle (direction), use trigonometry: \( \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} \).
2. Scale Drawing (for any angle):
If the vectors aren't at a right angle, you can draw them accurately using a ruler and a protractor.
Step-by-step:
• Choose a scale (e.g., 1 cm = 1 N).
• Draw the first vector at the correct angle.
• Draw the second vector "tip-to-tail."
• Measure the length of the resultant with a ruler and convert it back to the units (N, m/s, etc.).
• Measure the angle with a protractor.
Key Takeaway: The resultant is the shortcut from the very start to the very end of your vector journey.
4. Resolving Vectors into Components
Sometimes in Physics, we want to do the opposite of adding. We have a diagonal vector, and we want to know how much it is pushing horizontally (\( x \)) and how much it is pushing verticaly (\( y \)). This is called resolving a vector.
Imagine pulling a suitcase at an angle. Some of your force pulls the suitcase forward, and some of it lifts it up.
The Formulas
If you have a force \( F \) at an angle \( \theta \) to the horizontal:
• Horizontal component: \( F_x = F \cos \theta \)
• Vertical component: \( F_y = F \sin \theta \)
Memory Trick: "Cos is a-cross" the angle. If you have to move "across" the angle to get to the component, use Cos. The other one will be Sin.
Step-by-Step Example:
A kid pulls a sledge with a force of 50 N at an angle of 30° to the ground.
1. Horizontal force: \( 50 \times \cos(30) = 43.3 \) N.
2. Vertical force: \( 50 \times \sin(30) = 25.0 \) N.
Quick Review Box:
• Resolving = Splitting one diagonal vector into two perpendicular (90°) ones.
• Use \( F \cos \theta \) for the side adjacent to the angle.
• Use \( F \sin \theta \) for the side opposite the angle.
Key Takeaway: Resolving vectors allows us to simplify complex 2D problems into two simple 1D problems!