Formula & Dimensional Analysis
Every formula with derivation context, dimensional verification, and exam application notes.
🧮 Projectile Motion Calculator
Components of Initial Velocity
For θ = 45°: vₓ = vy = v₀/√2. For θ = 30°: vₓ = v₀√3/2, vy = v₀/2. Memorize these — they appear in 60% of projectile MCQs.
Position at Time t
These are the parametric equations of the parabolic trajectory. Eliminate t to get the trajectory equation.
Forgetting the negative sign in y equation. Always: y = vy₀t minus ½gt² (gravity opposes upward motion).
Velocity at Time t
Time of Flight (T)
Maximum Height (H)
Horizontal Range (R)
R = 4H·cotθ. Also: R = 4H when θ = 45°. JEE often gives H and asks R — use this directly.
Equation of Trajectory
The second form y = x tanθ(1 − x/R) is extremely useful when range R is known. Directly substitute x and find y without calculator.
Horizontal Projectile from Height h
🧮 Circular Motion Calculator
Angular Velocity
ω = 2πf — angular velocity increases with frequency. More rotations per second = faster spinning.
Centripetal Acceleration
Centripetal Force
Centripetal force is NOT a separate force. It is the NET force. For a car on a circular road, friction IS the centripetal force. For a satellite, gravity IS the centripetal force.
Time Period & Frequency
Non-Uniform Circular Motion — Net Acceleration
For non-uniform circular motion, the acceleration is not toward the center. JEE tests: at what angle is the net acc to the radius? Answer: tan⁻¹(aₜ/aᶜ).
Relative Velocity
Always work from: "velocity of what, w.r.t whom?" A w.r.t B = v_A minus v_B. NEVER add. Direction matters.
Magnitude of Relative Velocity
Special cases:
→ Same direction (α=0°): |v_AB| = |v_A − v_B|
→ Opposite (α=180°): |v_AB| = v_A + v_B
→ Perpendicular (α=90°): |v_AB| = √(v_A² + v_B²)
River-Boat: Minimum Time Path
River-Boat: Shortest Path
In JEE, if you derive a formula and want to verify it quickly — use dimensional analysis. Also, JEE sometimes gives a formula with unknown powers and asks you to find them using dimensional analysis. This appears in ~1 question per year.
| Quantity | Formula | Dimension | SI Unit |
|---|---|---|---|
| Position / Displacement | r⃗ = xî + yĵ | [L] | m |
| Velocity | v = dr/dt | [LT⁻¹] | m/s |
| Acceleration | a = dv/dt | [LT⁻²] | m/s² |
| Range (R) | v₀²sin2θ/g | [L] | m |
| Time of Flight (T) | 2v₀sinθ/g | [T] | s |
| Max Height (H) | v₀²sin²θ/2g | [L] | m |
| Angular Velocity (ω) | θ/t or v/r | [T⁻¹] | rad/s |
| Centripetal Acc (aᶜ) | v²/r or ω²r | [LT⁻²] | m/s² |
| Centripetal Force (F) | mv²/r | [MLT⁻²] | N |
| Time Period (T) | 2π/ω | [T] | s |
Dimensional Verification Examples
Verify: R = v₀²sin2θ/g
▼LHS: [R] = [L]
RHS: [v₀²] = [L²T⁻²], sin2θ is dimensionless, [g] = [LT⁻²]
[v₀²/g] = [L²T⁻²] / [LT⁻²] = [L] ✓
✅ Dimensionally consistent
Verify: aᶜ = v²/r
▼LHS: [aᶜ] = [LT⁻²]
RHS: [v²/r] = [L²T⁻²] / [L] = [LT⁻²] ✓
✅ Dimensionally consistent
✗ Tell you the correct formula if multiple formulas have same dimensions.
✗ Determine dimensionless constants (like 2, π, sin, cos, etc.).
✗ Distinguish between scalar and vector quantities.