↗️ Motion in a Plane

1. Scalars and Vectors

↗️ Vector Quantities

A quantity that has both magnitude and direction, and obeys the triangle law of addition. Examples: Displacement, Velocity, Force.

Vector Addition: Uses the Parallelogram Law. If two vectors A and B are at an angle θ, the resultant R is:

R = √(A² + B² + 2AB cosθ) Direction: tan α = (B sinθ) / (A + B cosθ)

2. Dot and Cross Products

Dot (Scalar) Product: A · B = AB cosθ. Gives a scalar. Used to find work done (W = F · s).
Cross (Vector) Product: A × B = AB sinθ n̂. Gives a vector. Used to find torque (τ = r × F).

3. Projectile Motion

When an object is thrown into the air with some initial velocity at an angle to the horizontal, it moves under the influence of gravity alone in a curved path (parabola).

Key Formulas for Projectile (Initial velocity 'u', angle 'θ')

1. Time of Flight (T): T = (2u sinθ) / g

2. Maximum Height (H): H = (u² sin²θ) / 2g

3. Horizontal Range (R): R = (u² sin 2θ) / g

Note: Maximum range occurs when θ = 45°.

4. Uniform Circular Motion

When a particle moves in a circle with constant speed, its velocity direction changes continuously, meaning it is accelerating.

Centripetal Acceleration (ac) = v² / r = ω²r Directed towards the center of the circle.