Kinematics
Complete Formula Sheet & Shortcut Bible · BITSAT 2026
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Core Vector Operations
Resultant Vector (R)
R = √(A² + B² + 2ABcosθ)
Dot Product (Scalar)
A · B = |A||B|cosθ
A · B = AₓBₓ + AᵧBᵧ + A₂B₂
A · B = AₓBₓ + AᵧBᵧ + A₂B₂
Cross Product (Vector)
|A × B| = |A||B|sinθ
Direction: Right-Hand Rule
Direction: Right-Hand Rule
Calculus in Kinematics
Position Vector (r⃗)
r⃗ = xî + yĵ + z k̂
Instantaneous Velocity (v⃗)
v⃗ = dr⃗/dt
Instantaneous Acceleration (a⃗)
a⃗ = dv⃗/dt = d²r⃗/dt²
Displacement from Velocity
Δr⃗ = ∫v⃗(t) dt
Velocity from Acceleration
Δv⃗ = ∫a⃗(t) dt
Equations of Motion (Constant Acceleration)
First Equation
v = u + at
Second Equation
s = ut + ½at²
Third Equation
v² = u² + 2as
Displacement in nᵗʰ second
sₙ = u + (a/2)(2n - 1)
BITSAT Speed Hacks
For motion under gravity, use `a = -g` for upward motion and `a = +g` for downward motion.
If starting from rest (`u=0`), ratio of distances in equal time intervals is 1:3:5:7... (Galileo's Law).
For a body dropped from height `h`, time to reach ground is `t = √(2h/g)` and final velocity is `v = √(2gh)`.
Time of ascent equals time of descent for vertical motion (neglecting air resistance).
Projectile Motion (Ground-to-Ground)
Time of Flight (T)
T = (2u sinθ) / g
Maximum Height (H)
H = (u² sin²θ) / 2g
Horizontal Range (R)
R = (u² sin2θ) / g
Equation of Trajectory
y = x tanθ - (gx²) / (2u² cos²θ)
Max Range Condition
θ = 45° → Rₘₐₓ = u²/g
Same Range Condition
Range is same for angles
θ and (90° - θ)
θ and (90° - θ)
Uniform Circular Motion (UCM)
Angular Velocity (ω)
ω = Δθ/Δt = 2π/T = 2πf
Linear Velocity (v)
v = ωr
v is tangent to path
v is tangent to path
Centripetal Acceleration (a꜀)
a꜀ = v²/r = ω²r
a꜀ is towards center
a꜀ is towards center
Centripetal Force (F꜀)
F꜀ = ma꜀ = mv²/r
Angular Acceleration (α)
α = dω/dt
(Zero for UCM)
(Zero for UCM)
Relative Motion
Relative Velocity
v⃗ₐᵦ = v⃗ₐ - v⃗ᵦ
(Velocity of A w.r.t. B)
(Velocity of A w.r.t. B)
Relative Acceleration
a⃗ₐᵦ = a⃗ₐ - a⃗ᵦ
River-Boat: Shortest Time
Swim perpendicular to river flow.
tₘᵢₙ = d / vₛᵣ
tₘᵢₙ = d / vₛᵣ
vₛᵣ is swimmer's speed in still water. d is river width.
River-Boat: Shortest Path
Swim upstream at angle θ.
sinθ = vᵣ / vₛᵣ
sinθ = vᵣ / vₛᵣ
vᵣ is river speed. Resultant velocity is perpendicular to flow.
Motion Type Comparison
| Parameter | 1D Motion (Linear) | 2D Motion (Projectile) | Uniform Circular Motion |
|---|---|---|---|
Velocity | Changes in magnitude | Both magnitude & direction change | Magnitude constant, direction changes |
Acceleration | Constant (e.g., `g`) | Constant (`aₓ=0`, `aᵧ=-g`) | Magnitude constant, direction changes (always radial) |
Path | Straight line | Parabolic | Circular |
Key Idea | Use 3 equations of motion | Separate x and y components | Centripetal force `mv²/r` is required |