Exams › JEE Advanced › Physics
Correct answer: 330/338 * f0
Phase 1 (man running on plank): Let vₘ = man's velocity (ground), vₚ = plank velocity (ground). vₘ - vₚ = 8 m/s (relative). Momentum conservation: 50*vₘ + 150*vₚ = 0. From these: 50*(vₚ+8) + 150*vₚ = 0 => 200*vₚ = -400 => vₚ = -2 m/s; vₘ = 6 m/s. Phase 2 (man jumps off same vₘ = 6 m/s): Momentum conservation: 50*6 + 150*vₚ_new = 50*6 + 150*(-2) = 300 - 300 = 0. Wait: before jump, total momentum = 50*6 + 150*(-2) = 300 - 300 = 0 (consistent). After jump, man has velocity 6 m/s (same), so 50*6 + 150*vₚ_new = 0 => vₚ_new = -300/150 = -2 m/s. Hmm, no change. Let me reconsider — while running, plank is at -2 m/s. When man jumps off (leaves contact), system momentum is still 0. Man goes at 6 m/s, plank at -2 m/s — but now there's no friction, so plank stays at -2 m/s and man stays at 6 m/s. After jumping: man velocity = +6 m/s, plank velocity = -2 m/s. Doppler: source = man moving at +6 m/s, observer = detector on plank moving at -2 m/s. Sound in still medium at 330 m/s. Taking rightward as positive. f_observed = f0 * (v_sound + v_observer) / (v_sound + v_source)... wait, need careful sign convention. If man moves right at 6 m/s and plank moves left at 2 m/s, they move apart. f = f0 * (v + v_D) / (v + v_S) where directions are measured as moving-toward-source positive. Observer moves toward source means leftward... no, observer (detector) moves at -2 m/s (left), source (man) moves at +6 m/s (right). They move apart. Standard Doppler: f = f0 * (v - v_observer) / (v - v_source)... using the formula where positive direction is from source toward observer. Actually: f = f0 * (v + vₒ)/(v + vₛ) where vₒ is speed of observer toward source and vₛ is speed of source toward observer. Observer moves AWAY from source at 2 m/s (leftward = away from right-moving man), so vₒ = -2 (away). Source moves AWAY from observer at 6 m/s. So f = f0*(330 - 2)/(330 + 6) = 328/336 * f0. But that gives option C. Let me recheck: Taking +x to the right. Man is at some point moving right at +6. Detector on plank moving at -2. Sound travels from man (source) to detector (observer). The component of observer velocity toward source: observer velocity = -2 (left), source is to the right of observer (man ran to the right), so toward source = +x direction. Observer's velocity component toward source = -2 (negative = moving away). Source velocity toward observer: source moves right (+6), observer is to the left, so toward observer = -x direction... source velocity toward observer = -6 (source moves away from observer too). f = f0*(v + v_obs_toward_source)/(v + v_src_away_from_obs)... using: f = f0*(v + v_O)/(v + v_S) where v_O = component of observer toward source (negative if away) and v_S = component of source toward observer (negative if receding). Actually standard formula: f_obs = f_source * (v_sound + v_observer) / (v_sound + v_source) where both v_observer and v_source are SIGNED with positive = toward each other... Let me use the most basic form: f = f0 * (v +/- vₒ) / (v -/+ vₛ). Observer moving away from source: use minus in numerator. Source moving away from observer: use plus in denominator. f = f0 * (330 - 2)/(330 + 6) = 328/336 * f0. That is option C. But wait — I need to reconsider phase 2. When the man 'jumps off', perhaps the problem means: initially they're both moving (man at 6, plank at -2), and when man jumps off and SLIDES on the surface (no friction between man and surface), the plank has NO force on it either (smooth surface), so plank continues at -2, man continues at 6. The Doppler analysis above gives 328/336. Let me check option D: 330/338. That would imply 330-2=328... no. 330/338 means something like (330)/(330+8). Hmm — maybe I made an error in computing plank velocity while man runs. Actually let me reconsider: while man is ON plank, man moves at +6 m/s (ground), plank at -2 m/s (ground). Man's velocity w.r.t. plank = 6-(-2) = 8 m/s. Good. When man jumps off: the instant he leaves, the plank no longer has a running person. The total momentum is 50*6 + 150*(-2) = 0. After man leaves at 6 m/s (ground), plank recoils. But momentum must still be 0: 50*6 + 150*v_plank = 0 => v_plank = -2 m/s. So plank doesn't change speed. Hence Doppler: f = f0*(330-2)/(330+6) = 328/336. Answer is option C.