Motion in a Plane & Projectile: JEE Main Guide

Motion in a Plane is among the first chapters students study and among the last they master. The equations look simple, but JEE frames projectile problems in oblique launches, river-crossing scenarios, and relative velocity setups that trip up students who only drilled the standard textbook derivations. This guide covers the full range of techniques — from the three basic projectile formulae to the cross-river minimum-time and minimum-drift strategies.

Projectile Motion: The Essential Formulae

A projectile launched at speed u at angle θ has horizontal component uₓ = u cosθ (constant) and vertical component u_y = u sinθ (affected by gravity). Time of flight: T = 2u sinθ/g. Range: R = u² sin2θ/g. Maximum height: H = u² sin²θ/(2g). Range is maximum at θ = 45°, giving R_max = u²/g. At complementary angles (θ and 90°−θ), the range is the same but H and T differ. JEE exploits this equal-range symmetry in questions asking for two possible angles of projection for a given range.

The trajectory equation: y = x tanθ − gx²/(2u² cos²θ). This is a downward-opening parabola. JEE occasionally gives the trajectory equation and asks for u, θ, or R. Identify coefficients of x and x² to extract tanθ and g/(2u² cos²θ). Projectile from a height: if launched horizontally from height H, T = √(2H/g) and horizontal range R = u√(2H/g). Take a free kinematics mock test to test your formula recall speed. For related work, see our kinematics complete guide.

Relative Velocity and River-Crossing Problems

Relative velocity v_AB = v_A − v_B. For two objects moving at angle θ between their velocities: |v_rel| = √(v_A² + v_B² − 2v_Av_B cosθ). Rain-man problems: a man moving with velocity v_m in rain falling at v_r must tilt his umbrella at angle α from vertical given by tan α = v_m/v_r — this is the direction of relative velocity of rain with respect to man. JEE sets up these problems both for rain and for aircraft in wind.

River crossing: boat speed v_b in still water, river current v_r perpendicular to desired path. For minimum time: head perpendicular to bank, drift = v_r × t_min where t_min = d/v_b. For minimum drift (zero if v_b > v_r): angle α = sin⁻¹(v_r/v_b) upstream, drift = 0, time = d/√(v_b² − v_r²). If v_b < v_r, minimum drift is achieved at a specific angle: derive using calculus (d(drift)/dα = 0). For circular motion fundamentals, see our circular motion guide.

Projectile on an Inclined Plane

When a projectile is launched from or onto an inclined plane of angle β, align the coordinate system with the incline (x along incline, y perpendicular). New gravitational components: g_x = g sinβ (decelerating along incline), g_y = g cosβ (perpendicular deceleration). Reapply all projectile formulae with these modified gravity components. Time of flight: T = 2u sinα/(g cosβ) where α is the angle of launch measured from the incline. Range along incline: R = 2u² sinα cos(α+β)/(g cos²β). JEE uses this for maximum range on an incline — differentiate with respect to α and set to zero.

The angular momentum of a projectile about the point of projection at time t: L = muₓ × (vertical drop) = m·u cosθ·(½gt²) — it increases with t². At the highest point, L = m·u cosθ·H. These are occasionally tested as conceptual "which quantity is conserved or changes" questions. For a broader overview of how Mechanics chapters interconnect for JEE, see our mechanics master guide.

Exam Mistakes and How to Avoid Them

The three most common errors in this chapter: (1) treating the initial vertical velocity as g (it is u sinθ); (2) forgetting that horizontal velocity is constant throughout flight; (3) applying river-crossing formulae for minimum time when minimum drift is asked (they require different angles). Before every projectile problem, write down uₓ = u cosθ, u_y = u sinθ, and a_y = −g — this 3-second habit prevents 90% of errors. For additional problem-types with inclined planes and projectiles see our friction and incline guide.