First slide

Third law

Question

An object flying in air with velocity $(20 \hat{i} + 25 \hat{j} - 12 \hat{k})$ suddenly breaks into two pieces whose masses are in the ratio 1 : 5. The smaller mass flies off with a velocity $(100 \hat{i} + 35 \hat{j} + 8 \hat{k})$ . The velocity of the larger piece will be,

Moderate

A

$- 20 \hat{i} - 15 \hat{j} - 80 \hat{k}$
B

$4 \hat{i} + 23 \hat{j} - 16 \hat{k}$
C

$- 100 \hat{i} - 35 \hat{j} - 8 \hat{k}$
D

$20 \hat{i} + 15 \hat{j} - 80 \hat{k}$

Solution

By conservation of linear momentum
${\vec{P}}_{i} = {\vec{P}}_{f}$
$m {\vec{v}}_{i} = (\frac{m}{6} {\vec{v}}_{1} + \frac{5 m}{6} {\vec{v}}_{2})$
${\vec{v}}_{i} = (\frac{{\vec{v}}_{1}}{6} + \frac{5}{6} {\vec{v}}_{2}) 20 \hat{i} + 25 \hat{j} - 12 \hat{k} = \frac{(100 \hat{i} + 35 \hat{j} + 8 \hat{k})}{6} + \frac{5 {\vec{v}}_{2}}{6} 120 \hat{i} + 150 \hat{j} - 72 \hat{k} = 100 \hat{i} + 35 \hat{j} + 8 \hat{k} + 5 {\vec{v}}_{2} 20 \hat{i} + 115 \hat{j} - 60 \hat{k} = 5 {\vec{v}}_{2} {\vec{v}}_{2} = \frac{20 \hat{i} + 115 \hat{j} - 80 \hat{k}}{5} {\vec{v}}_{2} = 4 \hat{i} + 23 \hat{j} - 16 \hat{k}$

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