Kinematics

Concept overview

Kinematics describes motion without asking about the cause. The three fundamental variables — position, velocity and acceleration — are related by calculus (v = dx/dt, a = dv/dt) and, for the special case of constant acceleration, by the three familiar equations of motion.

Motion in one dimension is the simplest testbed. Given any two of position, velocity, acceleration and time, the equations v = u + at, s = ut + ½at², and v² = u² + 2as give you the rest. The trap most students fall into is applying these equations to non-constant acceleration problems where they don't hold — always check that a is constant first.

Motion in two dimensions splits into independent horizontal and vertical components. This is the key insight of projectile motion: the horizontal velocity is constant (no horizontal force), while the vertical component undergoes uniform downward acceleration g. Range R = u² sin(2θ) / g and max height H = u² sin²θ / (2g) follow directly from combining these two independent motions.

Relative velocity extends the framework — the velocity of A with respect to B is just v_A − v_B, a vector subtraction. Boat-in-a-river and man-crossing-the-rain problems are standard applications and appear frequently in JEE Main.

Key formulas & concepts

• First equation of motion: v = u + at
• Second equation of motion: s = ut + ½at²
• Third equation of motion: v² = u² + 2as
• Projectile range: R = u² sin(2θ) / g
• Projectile max height: H = u² sin²θ / (2g)
• Time of flight: T = 2u sinθ / g
• Relative velocity: v_AB = v_A − v_B (vector)

Common mistakes to avoid

• Applying equations of motion to a problem where acceleration is not constant.
• Mixing scalar and vector quantities — displacement is a vector; distance is a scalar. They differ when the path is curved or reverses.
• Forgetting that acceleration due to gravity is downward — the vertical component of velocity decreases going up and increases going down.
• Solving projectile problems in coupled x-y variables instead of decoupling into two independent 1D problems.

JEE Main relevance

JEE Main asks 1–2 kinematics questions every paper — usually one projectile motion or relative velocity numerical, and one v-t or x-t graph interpretation. The chapter has a small formula set and clean archetypes.

JEE Advanced relevance

Advanced kinematics problems introduce non-constant acceleration (e.g., a(t) or a(x)) requiring integration. Vector-based relative-velocity problems in 2D with two moving objects (river-boat + wind) are also standard Advanced patterns.

Previous-year questions (PYQ)

Kinematics PYQs cluster into three archetypes: 1D constant acceleration, 2D projectile, and relative velocity. Drill each separately in the JEETutorAI chapter-wise practice bank.