Calculate the ratio of momentum,when
$(i)$ velocity of an object is doubled.
$(ii)$ mass of an object is halved.
$(iii)$ both mass and velocity are increased by three times.

(N/A) Let a body of mass $m$ be moving with velocity $v$. The initial momentum is $p_{1} = m v$.
$(i)$ When the velocity is doubled,the new velocity is $2v$. The new momentum $p_{2} = m(2v) = 2mv = 2p_{1}$. Therefore,the ratio $p_{1} : p_{2} = 1 : 2$.
$(ii)$ When the mass is halved,the new mass is $\frac{1}{2}m$. The new momentum $p_{3} = (\frac{1}{2}m)v = \frac{1}{2}mv = \frac{1}{2}p_{1}$. Therefore,the ratio $p_{1} : p_{3} = 2 : 1$.
$(iii)$ When both mass and velocity are increased by three times,the new mass is $3m$ and the new velocity is $3v$. The new momentum $p_{4} = (3m)(3v) = 9mv = 9p_{1}$. Therefore,the ratio $p_{1} : p_{4} = 1 : 9$.