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Chapter 03 · Part I · Mechanics
Revised on ____________________
Chapter 03 · Vectors & Motion in a Plane

Two dimensions, one pair of independent axes

2 sheets
2 diagrams
29 results

Formulas and conditions only — no derivations, no solved numbers. The diagrams are the reader's own.

I

Vector algebra

RESOLVE FIRST, THEN NEVER MIX THE AXES
I.Components  Ax = A cos θ , Ay = A sin θ
II.Magnitude  A = √(Ax² + Ay²)
III.Resultant  R = √(A² + B² + 2AB cos θ)
IV.Its direction  tan β = B sin θA + B cos θ measured from A
V.Dot product  A·B = AB cos θ scalar; work, flux, power
VI.Cross product  |A×B| = AB sin θ vector; torque, L, F on a charge
Two vectors of equal magnitude A give a resultant 2A cos(θ/2) — the fastest line in the paper, and it removes the cosine rule entirely.
II

Projectile from level ground

HORIZONTAL a = 0, VERTICAL a = −g
I.Time of flight  T = 2u sin θg
II.Maximum height  H = u² sin²θ2g
III.Range  R = u² sin 2θg
IV.Maximum range  Rmax = u²g at θ = 45°
V.Trajectory  y = x tan θ − g x²2u² cos²θ a parabola, always
VI.Velocity at time t  vx = u cos θ , vy = u sin θ − g t
VII.H and R together  R = 4Htan θ
VIII.Complementary angles  θ and (90° − θ) give the same R different T and H
xyno gravityv0θHR
III

Projectiles that do not start or land level

SAME PHYSICS, NEW AXES
I.Horizontal throw from h  t = √( 2hg ) , R = u t
II.Landing speed  v = √(u² + 2gh)
III.Landing angle  tan φ = √(2gh)u
IV.Up an incline α  R = 2u² sin(θ − α) cos θg cos²α θ measured from the horizontal
V.Its maximum  θ = α2 + 45° down-slope: use −α
VI.Landing ⊥ to a slope  the along-slope velocity is zero at impact
On an incline, rotate the axes to lie along and across the slope: gravity then has the two components g sin α and g cos α, and the problem becomes the level one again.
IV

Relative velocity in a plane

RIVERS, RAIN AND AIRCRAFT
I.Definition  vAB = vA − vB
II.Minimum time across  t = dvb aim straight across; you drift
III.That drift  x = vr dvb
IV.Zero drift  sin θ = vrvb needs vb > vr
V.Then the crossing time  t = d√(vb² − vr²)
VI.Rain on a moving man  tan θ = vmanvrain tilt the umbrella forward
BankBankvbvrActual(A) Min. timeBankBankvbvrvnet(B) Zero drift
V

Closest approach and collisions

SET THE RELATIVE PROBLEM UP ONCE
I.They collide if  vAB points along the initial separation
II.Closest approach  dmin = d sin φ φ between vAB and the line joining
III.Time to it  t = d cos φvAB
Where marks are lost in this chapter
1“Launched at 30° to the incline” means θ = 30° + α from the horizontal — not 30°. A favourite JEE trap.
2Adding magnitudes instead of vectors. |A + B| = A + B only when they are parallel.
3Using R = u² sin 2θ / g when the launch and landing points are at different heights. It does not apply.
4In river problems the boat's speed is given relative to the water, never to the bank.