Two particles of masses ${\mathrm{m}}_1$ and ${\mathrm{m}}_2$ are connected by a light and inextensible string which passes over a fixed pulley. Initially, the particle ${\mathrm{m}}_1$ moves with a velocity $v_0$ when the string is not taut. Neglecting friction in all contacting surfaces, find the velocities of the particles ${\mathrm{m}}_1$ and ${\mathrm{m}}_2$ just after the string is taut.

A plate of mass $M$ is held at rest by firing bullets from below. Each bullet has a mass $\mathrm{m}.$ velocity $u$ (up) just before hitting and stops after hitting the plate for a moment and falls. Determine the number of bullets striking the plate per unit time.

A particle of mass $1 \mathrm{kg}$ is attached to a string of length $5 \mathrm{m}.$ The string is attached to a fixed point $O.$ It is released from the position as shown in figure. Calculate

(a) the impulse developed in the string when it becomes taut;
(b) the velocity of the particle just after the string becomes taut;
(c) the impulse developed in this string $PQ$ at this instant.