A rectangular board (40 cm × 30 cm) moves at 2 cm s⁻¹ perpendicular to its 30 cm side. An insect traverses diagonal AC in 10 s relative to board, then returns at equal relative speed. If total ground-frame path length is a(6√5 + 2√13) cm, find a. [See Image.]
Physics · Mixed Chapter
JEE Main Practice

Question

A rectangular board (40 cm × 30 cm) moves at 2 cm s⁻¹ perpendicular to its 30 cm side. An insect traverses diagonal AC in 10 s relative to board, then returns at equal relative speed. If total ground-frame path length is a(6√5 + 2√13) cm, find a. [See Image.]

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1 — Relative insect speed is diagonal length /10 s = √(40²+30²)/10 = 5 cm s⁻¹. Relative velocity along A→C has components (4,3) cm s⁻¹. Step 2 — The board moves at (2,0) cm s⁻¹, perpendicular to its shorter (vertical) side. Ground-frame insect velocity on A→C is (6,3), of magnitude √45=3√5 cm s⁻¹. Step 3 — During return C→A, insect velocity relative board is (−4,−3), so ground velocity is (−2,−3), magnitude √13 cm s⁻¹. Step 4 — Each leg lasts 10 seconds: total distance = 10(3√5+√13) = 5(6√…