A coordinate axis system taking $x$ -axis as horizontal smooth floor is shown in figure. Two small balls of masses $m$ and $3 \mathrm{m}$ attached with a string are released from some heights on $y$ -axis as shown in figure. The balls may collide head on or obliquely. After a certain time mass $\mathrm{m}$ is at $(9 \mathrm{cm}, 20 \mathrm{cm})$ while mass $3 \mathrm{m}$ is $25 \mathrm{cm}$ above the $x$ axis and the string is taut. The balls always remain in $x-y$ plane. Find the length of string (in $\mathrm{cm}$)

A boy is playing carom board. Strikers is moving with velocity $u(=10 \mathrm{m}/\mathrm{s})$ and queen $Q$ is at rest. See angle $\theta \left(=37°\right)$ in the figure. Mass of $S$ and $Q$ are $m$ and $2 \mathrm{m}.$ Friction is absent and coefficient of resistution is $e.$ If after collision $Q$ moves along $SQ$ direction and $S$ moves along perpendicular to $SQ.$ Find the value of coefficient of restitution.

Mass of a particle and a hemisphere are $m$ and $3 \mathrm{m}/2$ respectively. Particle hits the hemisphere (at rest on smooth surface) with a horizontal velocity $v_0$ and rebounds in vertical direction after collision (neglect friction), if the value of $\theta$ (in degree) (collision is elastic):

The centres of the spheres $1, 2$ and $3$ lie on a single straight line. Sphere $1$ is moving with an (initial) velocity $v_1$ directed along this line and hits sphere $2$. Sphere $2$ acquires a velocity $v_2$ after collision, it hits sphere $3$. Both collisions are perfectly elastic. What must be the mass of sphere $2 (\mathrm{in} \mathrm{kg})$ for the sphere $3$ to acquire maximum velocity (The masses $m_1$ and $m_3$ of spheres $1$ and $3$ are $9 \mathrm{kg}$ and $1 \mathrm{kg}$ respectively).