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A particle is projected from the ground with initial velocity v0 = a*i + b*j (y-axis is vertical, x-axis is horizontal). It returns to the ground. Match each quantity in List-I with its correct expression in List-II. List-I: (P) Vx / Vy at time t = b / (2g) (Q) Maximum height reached (h_max) (R) Horizontal range (R) (S) |v_avg| / |v0| where v_avg is the average velocity over the entire flight List-II: (1) a / sqrt(a² + b²) (2) 2a / b (3) b² / (2g) (4) 2ab / g

1. P -> 1; Q -> 3; R -> 2; S -> 4
2. P -> 2; Q -> 3; R -> 4; S -> 1
3. P -> 2; Q -> 4; R -> 1; S -> 3
4. P -> 3; Q -> 4; R -> 1; S -> 2

Correct answer: P -> 2; Q -> 3; R -> 4; S -> 1

Solution

Initial velocity: vx = a (constant), vy0 = b. (P) At t = b/(2g): Vy = b - g*b/(2g) = b/2; Vx = a. Ratio Vx/Vy = a / (b/2) = 2a/b => matches (2). (Q) h_max = vy0²/(2g) = b²/(2g) => matches (3). (R) Total time T = 2b/g; R = a * T = 2ab/g => matches (4). (S) Displacement over full flight = (R, 0) = (2ab/g, 0); total time T = 2b/g; v_avg = (2ab/g) / (2b/g) = a; |v_avg|/|v0| = a/sqrt(a²+b²) => matches (1). Answer: P->2, Q->3, R->4, S->1.

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