Question

A body of mass m moving with a velocity v in the x-direction collides and sticks with another body of mass M moving with a velocity V in the y-direction. The magnitude of the momentum of the composite body is

• A. $(mv+MV)$
• B. $(m+M)(v+V)$
• C. $\sqrt{{\left(mv\right)}^2+{\left(MV\right)}^2}$
• D. $(Mv+mV)$

Answer 1

The correct option is C

$\sqrt{{\left(mv\right)}^2+{\left(MV\right)}^2}$

Let v' be the velocity of the composite body at an angle $\theta$ with the x-axis as shown in the figure. Equating the initial and the final momentum along x and y-axis,
$mv=(m+M)v^{\prime }cos\theta =P^{\prime }cos\theta$ .... (i)
and $MV=(m+M)v^{\prime }sin\theta =P^{\prime }sin\theta$ .... (ii)
where $P^{\prime }=(m+M)v^{\prime }$ is the momentum of the composite body. Squaring and adding equations (i) and (ii),
$(P^{\prime }{)}^2={\left(mv\right)}^2+{\left(MV\right)}^2\Rightarrow P^{\prime }=\sqrt{{\left(mv\right)}^2+{\left(MV\right)}^2}$
Hence, the correct choice is (c).