First slide

Law of conservation of momentum and it's applications

Question

A body of mass 1 kg initially at rest, explodes and breaks into three fragments of masses in the ratio 1 : 1: 3. The two pieces of equal mass fly off perpendicular to each other with a speed of 15 ms^-1 each. What is the velocity of the heavier fragment?

Moderate

A

$10 \sqrt{2} {ms}^{- 1}$
B

$5 \sqrt{3} {ms}^{- 1}$
C

$10 \sqrt{3} {ms}^{- 1}$
D

$5 \sqrt{2} {ms}^{- 1}$

Solution

Conservation of momentum during explosion yields
$\begin{array}{l} \Rightarrow {\vec{P}}_{1} + {\vec{P}}_{2} + {\vec{P}}_{3} = 0 \Rightarrow |\vec{P_{3}}| = |{\vec{P}}_{1} + {\vec{P}}_{2}| \\ \Rightarrow |\overset{}{m_{3}} \vec{v_{3}}| = |m_{1} {\vec{v}}_{1} + m_{2} {\vec{v}}_{2}| \\ since {\vec{v}}_{1} ⊥ {\vec{v}}_{2} \\ \Rightarrow |\vec{v_{3}}| = \sqrt{{(\frac{m_{1}}{m_{3}})}^{2} v_{1}^{2} + {(\frac{m_{2}}{m_{3}})}^{2} v_{2}^{2}} \\ \Rightarrow \sqrt{{(\frac{1}{3})}^{2} (15)^{2} + {(\frac{1}{3})}^{2} (15)^{2}} (∴ m_{1} : m_{2} : m_{3} = 1 : 1 : 3) \\ \Rightarrow v_{3} = 5 \sqrt{2} {ms}^{- 1} \end{array}$

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