A particle is moving along a vertical circle of radius $R$. At $P$, what will be the velocity of particle (assume critical condition at $C)$ ?

A bullet of mass $m$ and speed $v$ hits a pendulum bob of mass $M$ at time $t_1$, and passes completelythrough the bob. The bullet emerges at time $t_2$ with a speed of $v/2$. The pendulum bob is suspended by a stiff rod of length $l$ and negligible mass. After the collision, the bob can barely swing through a complete vertical circle.At time $t_3$, the bob reaches the highest position.What quantities are conserved in this process?

In the figure shown, a particle is released from the position $A$ on a smooth track. When the particle reaches at $B$, then normal reaction on it by the track is .........

A projectile is fired at a speed of $100\, m/sec$ at an angle of $37^o$ above the horizontal. At the highest point, the projectile breaks into two parts of mass ratio $1:3$ , the smaller coming to rest. Then the distance of heavier part from the launching point is ............... $\mathrm{m}$

A mass $M$ moving with a certain speed $V$ collides elastically with another stationary mass $m$. After the collision, the masses $M$ and $m$ move with speeds $V^{\prime}$ and $v$, respectively. All motion is in one dimension. Then,

A bullet weighing $10 \,g$ and moving with a velocity $300 \,m / s$ strikes a $5 \,kg$ block of ice and drop dead. The ice block is kept on smooth surface. The speed of the block after the collision is ........ $cm / s$