Motion in a Plane

Last chapter we moved only straight - forward/backward. Real life moves in plane also. Like a football kicked - it goes forward and up both. Like Swiggy boy taking turn at sector 18 - he changes direction. To describe this we need vectors, not just + and -. That's this chapter.

Official NCERT Chapter 3 keph103 = Motion in a Plane. Chapter 2 was Motion in a Straight Line. This is 2D motion.

Scalar: Only magnitude. Example - your water bottle mass 1kg, temperature 35°C, distance 10m. No direction needed. Add by normal maths.
Vector: Magnitude + direction + triangle law. Example - you walked 10m towards market (direction must). Displacement, velocity, acceleration, force are vectors.
Position vector: Where are you from origin? In plane, r = x i + y j. Like Google Maps - x = east, y = north. If you are at P(2,3), r = 2i + 3j.

Two steps walk: first 3m east (A), then 4m north (B). Total displacement? Not 7m! Because direction matters. Join head to tail - Triangle law: R = A+B. Magnitude by Pythagoras if perpendicular = 5m. Order doesn't matter: A+B = B+A commutative. (A+B)+C = A+(B+C) associative.
Parallelogram law: keep tails together, diagonal = sum.
Subtraction: A-B = A+(-B). -B means opposite direction.
Zero vector: A + (-A) = 0. Its direction not defined, magnitude 0. Happens when you return to start.

Any vector in plane can be broken into x and y. Like you pull suitcase at angle θ. Part goes horizontal, part vertical. Ax = A cosθ, Ay = A sinθ. Magnitude A = sqrt(Ax²+Ay²), tanθ = Ay/Ax. This is super imp for projectile. Any motion in plane = two independent 1D motions: x-motion and y-motion. They don't affect each other.

Position vector r(t) = x(t)i + y(t)j. Velocity v = dr/dt = vx i + vy j. Means vx = dx/dt, vy = dy/dt. Speed = |v| = sqrt(vx²+vy²). Acceleration a = dv/dt = ax i + ay j. So plane motion is just two straight line motions happening together.
If acceleration constant (a = constant), then equations like 1D but vector form:
v = v0 + a t
r = r0 + v0 t + ½ a t²
vx = v0x + ax t, x = x0 + v0x t + ½ ax t² (same for y)

In 2D, relative still V_AB = V_A - V_B vector subtraction. Classic is boat-river.
Boat velocity wrt water = Vb, water wrt ground = Vr, boat wrt ground = Vb + Vr vector sum.
Two cases: Shortest time: point boat directly across river (perpendicular), time = width / Vb. Drift = Vr × time.
Shortest path: point upstream at angle sinθ = Vr/Vb so that drift zero, boat goes straight across. Time longer = width / sqrt(Vb²-Vr²).

Any object thrown in air with only gravity acting (ignore air) is projectile. Path = parabola. Why? Horizontal velocity constant (no force), vertical accelerated by g down.
Throw with u at angle θ:
ux = u cosθ constant, uy = u sinθ changes.
Time of flight T = 2 u sinθ / g (time in air).
Max height H = u² sin²θ / 2g (at top vy=0).
Range R = u² sin2θ / g (total horizontal distance).
Trajectory y = x tanθ - (g x²)/(2 u² cos²θ) parabola equation.
Max range at θ=45° gives R_max = u²/g. Two angles that give same range: θ and 90°-θ (like 30° and 60°).

Motion in circle at constant speed. Speed constant but direction changes every moment, so velocity changes, so acceleration present. Acceleration towards centre called centripetal acceleration a_c = v²/r = ω² r. Towards centre always, even if speed constant.
Angular speed ω = v/r rad/s. Period T = 2πr/v = 2π/ω time for one round. Frequency f =1/T Hz.
Velocity tangent to circle, acceleration radial inward. They are perpendicular.