Scalars and vectors
Section: Physical Quantities and Units | Syllabus: Cambridge AS Level Physics (9702)
Scalars vs Vectors
All physical quantities can be classed as either scalar or vector quantities.
- Scalar quantities: Have magnitude (size) and a unit only. They are independent of direction.
- Vector quantities: Have both magnitude (with unit) and a particular direction.
| Scalar Quantities | Vector Quantities |
|---|---|
| Mass (kg) | Force / Weight / Friction (N) |
| Length / Distance (m) | Displacement (m) |
| Time (s) | Velocity (m s^-1) |
| Speed (m s^-1) | Acceleration (m s^-2) |
| Energy / Work Done (J) | Momentum (N s or kg m s^-1) |
| Power (W) | Electric Field (V m^-1) |
| Current (A) | Magnetic Flux Density (T) |
| Pressure (Pa) & Temp (K) | Gravitational Field (N kg^-1) |
- Energy: All forms of energy (even kinetic energy associated with movement) are scalars because they have no direction.
- Current: Electric current has direction in a circuit, but it is a scalar because it does not follow vector addition rules.
- Fields: Electric and magnetic fields are vectors (required for A Level).
Adding Collinear Vectors
Collinear vectors act along the same straight line. To add them:
- Choose a direction as positive.
- Assign a negative sign to vectors acting in the opposite direction.
- The sum is the resultant vector (the single vector that has the same effect as all original vectors combined).
Question: An aeroplane flies with a velocity of 250 m s^-1 relative to the air. The wind blows with a velocity of 40 m s^-1. Calculate the velocity relative to the ground if flying: (a) in the same direction, (b) in the opposite direction.
Solution(a) Same direction:
Velocity = 250 + 40 = 290 m s^-1 (in the direction of flight)(b) Opposite direction:
Velocity = 250 + (-40) = 210 m s^-1 (in the direction of flight) Physics TipA zero resultant force does not mean an object is stationary; it means its acceleration is zero (it could be moving at a constant velocity).
Adding Coplanar Vectors Using Scale Diagrams
Coplanar vectors act in the same flat plane. When they are not in a straight line, we use the Head-to-Tail method or the Parallelogram Rule.
Scale Diagram Placeholder
(Visualizing vectors drawn end-to-end to find a resultant R)
Worked Example: Graphical MethodQuestion: Two forces, 120 N and 50 N, act on a mass with an angle of 120° between them. Determine the magnitude and direction of the resultant.
Graphical Solution Steps- Scale: 1 cm = 20 N.
- Draw the 120 N force horizontally (6 cm).
- Use a protractor to draw the 50 N force (2.5 cm) at 120° from the first.
- Draw the resultant from the start of the first to the end of the second.
- Resultant Magnitude: Measurement of 5.1 cm × 20 = 102 N (approx).
- Direction: Angle measured relative to 120 N force = ~25°.
Perpendicular Vectors
When two vectors act at 90° to each other, the resultant is the hypotenuse of a right-angled triangle. We solve this using Pythagoras' Theorem and Trigonometry.
Magnitude: Resultant^2 = F_1^2 + F_2^2
Direction: tan θ = (Opposite/Adjacent)
Worked Example: Boat in a RiverQuestion: A boat travels at 2.5 m s^-1 directly across a river. The river flows at 0.92 m s^-1 perpendicular to the boat. Calculate the resultant velocity.
Solution1. Magnitude:
v = √(2.5^2 + 0.92^2) = √(6.25 + 0.8464) = √(7.0964) = 2.66 m s^-12. Direction (θ relative to path across river):
tan θ = (0.92/2.5) θ = tan^-1(0.368) = 20.2° away from the crossing path. Worked Example: Resultant ComponentsQuestion: An aeroplane flies North at 180 km h^-1. A side wind from the East acts at 90°. The resultant velocity is 187 km h^-1. Find the wind speed.
Solution 187^2 = 180^2 + (Wind Speed)^2 Wind Speed = √(187^2 - 180^2) = √(34969 - 32400) = √(2569) = 50.7 km h^-1Resolving Vectors into Components
Resolving a vector means splitting it into two perpendicular components (usually horizontal and vertical).
Horizontal component (along angle): F_x = F cos θ
Vertical component (opposite angle): F_y = F sin θ
Worked Example: Climbing AeroplaneQuestion: An aeroplane climbs at 10° to the horizontal with a constant velocity of 150 m s^-1. Find the horizontal and vertical velocity components.
Solution(a) Horizontal component:
v_x = 150 × cos(10°) = 150 × 0.985 = 148 m s^-1(b) Vertical component:
v_y = 150 × sin(10°) = 150 × 0.174 = 26 m s^-1 Worked Example: Horse Pulling a BoatQuestion: A horse pulls a boat along a canal using a rope at 15° to the forward direction. The force is 210 N. Calculate the forward component of the force.
Solution F_forward = 210 × cos(15°) = 210 × 0.966 = 203 NAdding Non-Perpendicular Coplanar Vectors Using Components
To add vectors at any angle:
- Resolve all vectors into x and y components.
- Sum the x-components (Σ F_x) and y-components (Σ F_y).
- Apply Pythagoras to obtain the final resultant magnitude (R = √(Σ F_x^2 + Σ F_y^2)).
Question: Two ropes pull a tree. Force A = 65 N at 20° to line X. Force B = 75 N at 15° to line X on the opposite side. Find the magnitude and direction of the resultant.
Solution1. Resolve into components (relative to line X):
- A_x = 65 cos 20° = 61.1 N; A_y = 65 sin 20° = 22.2 N
- B_x = 75 cos 15° = 72.4 N; B_y = -75 sin 15° = -19.4 N (opposite direction)
2. Sum components:
Σ F_x = 61.1 + 72.4 = 133.5 N Σ F_y = 22.2 - 19.4 = 2.8 N3. Resultant:
R = √(133.5^2 + 2.8^2) = 133.5 N θ = tan^-1((2.8/133.5)) = 1.2° from line X towards A. Worked Example: Bird Resultant VelocityQuestion: A bird flies 25 m s^-1 North. The wind blows at 14 m s^-1 at 38° from North. Find the resultant magnitude and direction.
Solution1. Components (X=East, Y=North):
- Bird: v_x = 0; v_y = 25
- Wind: v_x = 14 sin 38° = 8.62; v_y = 14 cos 38° = 11.03
2. Sum:
Σ v_x = 8.62; Σ v_y = 25 + 11.03 = 36.033. Resultant:
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