Ch 4 extends kinematics to two dimensions. Students learn vector algebra, resolve motion into components, study projectile motion, and analyse uniform circular motion.
Scalars: magnitude only (mass, time, temperature). Vectors: magnitude + direction (displacement, velocity, force). Addition: triangle law, parallelogram law. Resolution into components: A_x = A cosθ, A_y = A sinθ. Unit vectors: î, ĵ, k̂. Dot product: A·B = AB cosθ (scalar). Cross product: A×B = AB sinθ (vector).
Projectile: object moving under gravity only (after launch). Treat horizontal (constant velocity) and vertical (acceleration g) independently. For launch angle θ: Range R = u²sin2θ/g (max at 45°). Max height H = u²sin²θ/2g. Time of flight T = 2u sinθ/g. Path is parabolic.
Object moves in a circle with constant speed. Velocity direction changes continuously → centripetal acceleration a_c = v²/r = ω²r directed towards centre. Period T = 2πr/v = 2π/ω. Angular velocity ω = 2π/T. Examples: satellites, planets, cars on curves.
Download: https://ncert.nic.in/textbook/pdf/keph104.pdf | Part I: https://ncert.nic.in/textbook/pdf/keph1ps.zip
Range R = u²sin2θ/g. The range is maximum when sin2θ is maximum, i.e., sin2θ = 1, which gives 2θ = 90° → θ = 45°. At 45°, the horizontal and vertical components are equal, optimising the balance between distance and time in the air.
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