Learning Objectives
- Understand scalar and vector quantities and their operations
- Learn resolution of vectors and vector addition
- Study projectile motion and derive relevant equations
- Understand uniform circular motion and centripetal acceleration
- Apply vector concepts to two-dimensional kinematics
Key Concepts
Scalars and Vectors
Scalar: A quantity with only magnitude (e.g., mass, speed, temperature, energy).
Vector: A quantity with both magnitude and direction (e.g., displacement, velocity, force, momentum).
Unit Vector: A vector with magnitude 1, denoted by a hat symbol. â = A/|A|. The unit vectors along x, y, z axes are î, ĵ, k̂.
Vector Operations
Vector Addition (Triangle Law): Place vectors head to tail; the resultant goes from the tail of the first to the head of the last.
Parallelogram Law: If two vectors are represented as adjacent sides of a parallelogram, the diagonal gives the resultant.
Resultant magnitude: R = √(A² + B² + 2AB cos θ)
Direction: tan α = B sin θ / (A + B cos θ)
Vector Subtraction: A - B = A + (-B)
Resolution of Vectors
Any vector A in a plane can be resolved into two perpendicular components:
Aₓ = A cos θ (x-component), Aᵧ = A sin θ (y-component)
A = Aₓ î + Aᵧ ĵ, |A| = √(Aₓ² + Aᵧ²), tan θ = Aᵧ/Aₓ
Dot Product (Scalar Product)
A · B = AB cos θ = AₓBₓ + AᵧBᵧ + A_zB_z
Properties: Commutative, distributive. A · A = A². If A ⊥ B, then A · B = 0.
Cross Product (Vector Product)
A × B = AB sin θ n̂ (where n̂ is perpendicular to both A and B by right-hand rule)
Properties: Not commutative (A × B = -B × A), distributive. If A ∥ B, then A × B = 0.
î × ĵ = k̂, ĵ × k̂ = î, k̂ × î = ĵ
Projectile Motion
A projectile is any object thrown with an initial velocity that moves under gravity alone. The path is a parabola.
For a projectile launched with velocity u at angle θ with horizontal:
- Horizontal component: uₓ = u cos θ (remains constant)
- Vertical component: uᵧ = u sin θ (changes due to g)
- Time of flight: T = 2u sin θ / g
- Maximum height: H = u² sin²θ / 2g
- Range: R = u² sin 2θ / g
- Maximum range at θ = 45°: R_max = u²/g
- Equation of trajectory: y = x tan θ - gx²/(2u² cos²θ)
Complementary angles (θ and 90° - θ) give the same range.
Uniform Circular Motion
An object moving in a circle with constant speed has continuously changing velocity direction.
Angular velocity: ω = Δθ/Δt = 2π/T = 2πf (rad/s)
Linear speed: v = rω
Centripetal acceleration: a_c = v²/r = rω² (directed towards the centre)
Time period: T = 2πr/v = 2π/ω
Frequency: f = 1/T = ω/2π
Summary
Motion in a plane requires vector analysis. Vectors can be added using the triangle or parallelogram law and resolved into components. Projectile motion is a combination of uniform horizontal motion and uniformly accelerated vertical motion under gravity, resulting in a parabolic trajectory. Uniform circular motion involves constant speed but changing direction, requiring centripetal acceleration directed towards the centre.
Important Terms
- Resultant Vector: Single vector that produces the same effect as two or more vectors combined
- Resolution: Splitting a vector into perpendicular components
- Projectile: Object moving under the influence of gravity alone after being launched
- Trajectory: Path followed by a projectile (parabolic)
- Range: Horizontal distance covered by a projectile
- Centripetal Acceleration: Acceleration directed towards the centre in circular motion
Quick Revision
- R = √(A² + B² + 2AB cos θ) for resultant of two vectors
- Projectile: T = 2u sin θ/g, H = u² sin²θ/2g, R = u² sin 2θ/g
- Maximum range at θ = 45°
- Centripetal acceleration: a = v²/r = rω², directed toward centre
- v = rω, T = 2π/ω
- A · B = AB cos θ (scalar), A × B = AB sin θ n̂ (vector)