First slide

The steps in collision

Question

An object flying in air with velocity $(20 \hat{i} + 25 \hat{j} - 12 \hat{k})$ suddenly breaks in 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

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

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

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

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

Solution

Let m be the mass of an object flying with velocity v in air. When it gets split into two pieces of masses in ratio 1 : 5, the mass of smaller piece is m/6 and of bigger piece is $\frac{5 m}{6} .$
This situation can be interpreted diagrammatically as below.
As, the object breaks in two pieces, so the momentum of the system will remains conserved, i.e. the total momentum (before breaking) = total momentum (after breaking)
$m v = \frac{m}{6} v_{1} + \frac{5 m}{6} v_{2} \Rightarrow v = \frac{v_{1}}{6} + \frac{5 v_{2}}{6}$
Given, $v = 20 \hat{i} + 25 \hat{j} - 12 \hat{k}$
and $v_{1} = 100 \hat{i} + 35 \hat{j} + 8 \hat{k}$
Putting these values in Eq. (i), we get
$(20 \hat{i} + 25 \hat{j} - 12 \hat{k}) = \frac{(100 \hat{i} + 35 \hat{j} + 8 \hat{k})}{6} + \frac{5 v_{2}}{6}$
$\Rightarrow (120 \hat{i} + 150 \hat{j} - 72 \hat{k}) = (100 \hat{i} + 35 \hat{j} + 8 \hat{k}) + 5 v_{2}$
$\Rightarrow v_{2} = \frac{1}{5} (20 \hat{i} + 115 \hat{j} - 80 \hat{k})$
$= 4 \hat{i} + 23 \hat{j} - 16 \hat{k}$

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