A body of mass M at rest explodes into three | vedclass

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Question

(C) Let the mass of the body be $M$. Initially,the body is at rest,so the initial momentum is $P_i = 0$.After the explosion,the body splits into three

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$2.5$

Vedclass · Answer

(C) Let the mass of the body be $M$. Initially,the body is at rest,so the initial momentum is $P_i = 0$.After the explosion,the body splits into three pieces: $m_1 = M/4$,$m_2 = M/4$,and $m_3 = M - (M/4 + M/4) = M/2$.Let the velocities of the first two pieces be $\vec{v}_1 = 3\hat{i} \ m/s$ and $\vec{v}_2 = 4\hat{j} \ m/s$.The momentum of the first two pieces is $\vec{p}_1 + \vec{p}_2 = (M/4)(3\hat{i}) + (M/4)(4\hat{j}) = (3M/4)\hat{i} + (M)\hat{j}$.By the law of conservation of momentum,$\vec{p}_1 + \vec{p}_2 + \vec{p}_3 = 0$,so $\vec{p}_3 = -(\vec{p}_1 + \vec{p}_2) = -(3M/4)\hat{i} - (M)\hat{j}$.The velocity of the third piece $\vec{v}_3$ is given by $\vec{p}_3 / m_3 = [-(3M/4)\hat{i} - (M)\hat{j}] / (M/2) = -1.5\hat{i} - 2\hat{j} \ m/s$.The magnitude of the velocity is $v_3 = \sqrt{(-1.5)^2 + (-2)^2} = \sqrt{2.25 + 4} = \sqrt{6.25} = 2.5 \ m/s$.

$A$ body of mass $M$ at rest explodes into three pieces,two of which of mass $M/4$ each are thrown off in perpendicular directions with velocities of $3 \ m/s$ and $4 \ m/s$ respectively. The third piece will be thrown off with a velocity of.......... $m/s$.

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