Motion in Two Dimensions (Kinematics – 2D)
Relative velocity, swimmer problems, projectile motion & inclined plane (JEE Main essentials).
1) Relative Motion & Relative Velocity
- Motion analysed from a moving frame is called relative motion.
- Relative velocity of A w.r.t. B:
V⃗_AB = V⃗_A − V⃗_B - For 1D motion:
- Same direction: |V_AB| = |V_A − V_B|
- Opposite direction: |V_AB| = V_A + V_B
Relative velocity is always the vector difference of velocities.
2) Swimmer / Boat in River Problems
V⃗_s,g = V⃗_s,w + V⃗_w,g
- Minimum time to cross:
Swimmer should head perpendicular to river flow.
t_min = d / v - Reaching exactly opposite bank (only if v > u):
sinθ = u / v - Shortest path / minimum drift (v < u):
sinθ = v / u
3) Projectile Motion (Oblique Projection)
- Horizontal motion: aₓ = 0
- Vertical motion: aᵧ = −g
x = u cosθ · t
y = u sinθ · t − (1/2)gt²
Trajectory: y = x tanθ − (g x²) / (2u² cos²θ)
Time of flight: T = (2u sinθ) / g
Range: R = (u² sin2θ) / g
Maximum height: H = (u² sin²θ) / (2g)
For a given speed, the same range is obtained for angles θ and (90° − θ).
4) Velocity Direction & Special Results
- At highest point: vertical velocity = 0, horizontal velocity remains constant.
- Time of ascent = time of descent.
- Velocity becomes perpendicular to initial velocity at:
t = u / (g sinθ)
5) Projectile Motion on Inclined Plane (Angle α)
- Acceleration components:
aₓ = g sinα aᵧ = g cosα - Time of flight:
T = (2u sinθ) / (g cosα) - Maximum range (up the plane):
R_max = u² / [g(1 + sinα)] - Maximum range occurs when:
θ = 45° − α/2
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Last modified: December 14, 2025
