In the figure shown, the system is at rest, initially. Two persons, $ A $ and $ B $, of masses $ 40 \mathrm{~kg} $ each, move with speeds $ v_{1} $ and $ v_{2} $, respectively, towards each other on a plank lying on a smooth horizontal surface as shown in the figure. The plank travels a distance of $ 20 \mathrm{~m} $ towards right direction in $ 5 \mathrm{~s} $. (Here, $ v_{1} $ and $ v_{2} $ are given with respect to the plank). Then, (i) $ v_{1}=10 \mathrm{~m} \mathrm{~s}^{-1}, v_{2}=0 \mathrm{~m} \mathrm{~s}^{-1} $ (ii) $ v_{1}=15 \mathrm{~m} \mathrm{~s}^{-1}, v_{2}=5 \mathrm{~m} \mathrm{~s}^{-1} $ (iii) $ v_{1}=20 \mathrm{~m} \mathrm{~s}^{-1}, v_{2}=10 \mathrm{~m} \mathrm{~s}^{-1} $

Question

In the figure shown, the system is at rest, initially. Two persons, $ A $ and $ B $, of masses $ 40 \mathrm{~kg} $ each, move with speeds $ v_{1} $ and $ v_{2} $, respectively, towards each other on a plank lying on a smooth horizontal surface as shown in the figure. The plank travels a distance of $ 20 \mathrm{~m} $ towards right direction in $ 5 \mathrm{~s} $. (Here, $ v_{1} $ and $ v_{2} $ are given with respect to the plank). Then, (i) $ v_{1}=10 \mathrm{~m} \mathrm{~s}^{-1}, v_{2}=0 \mathrm{~m} \mathrm{~s}^{-1} $ (ii) $ v_{1}=15 \mathrm{~m} \mathrm{~s}^{-1}, v_{2}=5 \mathrm{~m} \mathrm{~s}^{-1} $ (iii) $ v_{1}=20 \mathrm{~m} \mathrm{~s}^{-1}, v_{2}=10 \mathrm{~m} \mathrm{~s}^{-1} $, only (i) and (iii) are possible., all (i), (ii), (iii) are not possible., only (ii) and (iii) are possible., All (i), (ii), (iii) are possible.

MARKS App · Accepted Answer

Let the distance travelled by plank be s in time t . So, the velocity of plank, v P = s t . ⇒ v P = 20 5 ⇒ v P = 4 m s - 1 ∆ x CM = 0 . Velocity of A with respect to ground = v 1 + 4 Velocity of B with respect to ground = v 2 - 4 As the net external force in x - direction is zero, the linear momentum is conserved. 40 v 1 + 4 - 40 v 2 - 4 + 20 × 4 = 0 . 40 v 1 - v 2 + 10 = 0 v 1 - v 2 = - 10 . If v 1 = 10 m s - 1 then v 2 = 20 m s - 1 If v 1 = 20 m s - 1 then v 2 = 30 m s - 1 If v 1 = 15 m s - 1 then v 2 = 25 m s - 1 .

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