MechanicsHigh Yield (2-3 Qs in JEE)

Kinematics & Motion in 1D & 2D Formulas

Uniformly accelerated motion, projectile motion, trajectory parabola, circular kinematics, and relative velocity.

8 Formulas•560+ Questions in Bank•

Concept Diagram

2D Projectile Trajectory

Parabolic trajectory showing launch velocity u at angle θ, max height H, horizontal range R, and time of flight.

v = u + at, \quad s = ut + \frac{1}{2}at^2, \quad v^2 = u^2 + 2as, \quad s_n = u + \frac{a}{2}(2n - 1)

Standard 1D and vector kinematic equations.

Examiner Tip: s_n is the displacement specifically in the n-th second.

T = \frac{2u\sin\theta}{g}, \quad H = \frac{u^2\sin^2\theta}{2g}, \quad R = \frac{u^2\sin 2\theta}{g}

Diagram Reference:Projectile Motion Core Parameters

Time of flight, maximum height, and horizontal range on flat ground.

Examiner Tip: Range is identical for complementary launch angles θ and (90° - θ).

y = x\tan\theta - \frac{gx^2}{2u^2\cos^2\theta} = x\tan\theta \left(1 - \frac{x}{R}\right)

Parabolic trajectory equation linking horizontal coordinate x to vertical coordinate y.

Examiner Tip: The form y = x tan θ (1 - x/R) saves massive calculation time when Range R is known.

a_c = \frac{v^2}{r} = \omega^2 r, \quad a_t = \frac{dv}{dt} = \alpha r, \quad a_{total} = \sqrt{a_c^2 + a_t^2}

Kinematics of circular motion with radial and tangential acceleration components.

Examiner Tip: Uniform circular motion has a_t = 0 (constant speed, changing velocity direction).

\vec{v}_{AB} = \vec{v}_A - \vec{v}_B, \quad t_{min} = \frac{d}{v_{br}}, \quad \text{Zero Drift Condition: } \sin\theta = \frac{v_r}{v_{br}} \; (v_{br} > v_r)

Relative velocity in 2D and crossing a river of width d.

Examiner Tip: For shortest time, boat must steer perpendicular to the river bank.

T = \frac{2u\sin(\alpha - \beta)}{g\cos\beta}, \quad R = \frac{u^2}{g\cos^2\beta}[\sin(2\alpha - \beta) - \sin\beta], \quad R_{\text{max}} = \frac{u^2}{g(1 + \sin\beta)}

Launch angle α from horizontal onto plane inclined at angle β. Replace g with components g cos β and g sin β.

Examiner Tip: Maximum range up the plane occurs when α = π/4 + β/2; down the plane replace β with -β.

\rho = \frac{v^2}{a_\perp} = \frac{[1 + (dy/dx)^2]^{3/2}}{|d^2y/dx^2|}, \quad \rho_{\text{top}} = \frac{u^2\cos^2\theta}{g}

Radius of the osculating circle at any point along a curved path where a_⊥ is normal acceleration.

Examiner Tip: At the highest point of standard projectile motion, speed is u cos θ and normal acceleration is g.

\vec{v}_{r/m} = \vec{v}_r - \vec{v}_m, \quad \tan\theta = \frac{|\vec{v}_m|}{|\vec{v}_r|}

Apparent direction of rainfall relative to a moving observer.

Examiner Tip: To protect from rain, hold umbrella in the direction of apparent velocity v_{r/m}.

Frequently tested misconceptions in Kinematics & Motion in 1D & 2D

•Applying kinematic equations (v = u + at) when acceleration is NOT constant.
•Forgetting that horizontal velocity component remains constant throughout projectile flight (u_x = u cos θ).