Particle $A$ of mass $m _{1}$ moving with velocity $(\sqrt{3} \hat{i}+\hat{j})\, ms ^{-1}$ collides with another particle $B$ of mass $m _{2}$ which is at rest initially. Let $\overrightarrow{ V }_{1}$ and $\overrightarrow{ V }_{2}$ be the velocities of particles $A$ and $B$ after collision respectively. If $m _{1}=2\, m _{2}$ and after collision $\overrightarrow{ V }_{1}=(\hat{ i }+\sqrt{3} \hat{ j })\, ms ^{-1},$ the angle between $\overrightarrow{ V }_{1}$ and $\overrightarrow{ V }_{2}$ is$......^o$

A man of $60\,kg$ is running on the road and suddenly jumps into a stationary trolly car of mass $120\,kg$. Then. the trolly car starts moving with velocity $2\,ms ^{-1}$. The velocity of the running man was _________$ms ^{-1}$. when he jumps into the car

A shell of mass $m$ moving with velocity $ v$ suddenly breaks into $2$ pieces. The part having mass $m/4$ remains stationary. The velocity of the other shell will be

A bullet of mass $50$ gram is fired from a $5 \,kg$ gun with a velocity of $1km/s$. the speed of recoil of the gun is .......... $m/s$

A stationary body of mass $m$ gets exploded in $3$ parts having mass in the ratio of $1 : 3 : 3$. Its two fractions having equal mass moving at right angle to each other with velocity of $15\,m/sec$. Then the velocity of the third body is

A hemisphere of radius $R$ and of mass $4m$ is free to slide with its base on a smooth horizontal table. A particle of mass $m$ is placed on the top of the hemisphere. The angular velocity of the particle relative to hemisphere at an angular displacement $\theta $ when velocity of hemisphere $v$ is