A long plank$\text{'}A\text{'}$ is at rest on a smooth horizontal surface. A small block ' $B$ ', whose mass is half of ' $A$ ', is placed on the plank at one end and projected along $A$ with some velocity $u.$ The coefficient of friction between the plank and block is $\mu$ :

A uniform wire frame $ABC$ is in the shape of an equilateral triangle in $xy$ -plane with centroid at the origin.

A molecule capable of vibrations is modeled as two point atoms of mass $M$ connected by a spring. Initially molecule is not vibrating when it collides elastically with a single atom of mass $M$ as shown in figure. Mark the CORRECT option(s).

Two blocks of mass ${\mathrm{m}}_1$ and ${\mathrm{m}}_2$ are connected by a spring in its natural length having spring constant $k$ and are placed on a smooth surface. Block ${\mathrm{m}}_1$ and ${\mathrm{m}}_2$ are given velocities $v$ and $v_2$ as shown in figure. Choose CORRECT statements.

A shell of mass $\mathrm{m}$ is fired from a cannon of mass $nm$ (without the shell) with a relative speed of $v_0.$ The shell is projected horizontally. At the time of firing.

A car ' $A$ ' of mass $\mathrm{m}$ is initially at rest on a plank of mass $M$ tied to the identical plank ' $C$ ' of same mass $M$ through a massless inextensible string as shown in the figure. The car accelerates from rest to velocity $v_0$ with respect to plank ' $B$ ' in time $t_0\sec .$ At time $t=t_0$ the car applies brake and comes to rest relative to plank in negligible time. Neglect friction between planks and ground.

Figure shows position and velocities of two particles moving under mutual gravitational attraction in space at time $t=0.$ The position of centre of mass after one second is '*' $\mathrm{m}.$ Fill '*'.