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Physics Question on work, energy and power

A mass mm moving horizontally (along the x−axis)x-axis) with velocity vv collides and sticks to a mass of 3m3m moving vertically upward (along the yy -axis) with velocity 2 v2\,v. The final velocity of the combination is

A

34vi^+14vj^\frac{3}{4} v \widehat{i} + \frac{1}{4} v \widehat{j}

B

14vi^+32vj^\frac{1}{4} v \widehat{i} + \frac{3}{2} v \widehat{j}

C

13vi^+23vj^\frac{1}{3} v \widehat{i} + \frac{2}{3} v \widehat{j}

D

23vi^+13vj^\frac{2}{3} v \widehat{i} + \frac{1}{3} v \widehat{j}

Answer

14vi^+32vj^\frac{1}{4} v \widehat{i} + \frac{3}{2} v \widehat{j}

Explanation

Solution

From the law of conservation of linear momentum
mv i^+(3m)(2 v)j^mv \,\hat{ i }+(3 m )(2\, v ) \hat{ j }
=4 mv′=4 \,mv '
v′=v4i^+32vj^v'=\frac{v}{4} \hat{i}+\frac{3}{2} v\hat{j}