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 '*'.

Three particles $A, B$ and $C$ of equal mass move with equal speed $v=5{ \mathrm{m} \mathrm{s}}^{-1}$ along the medians of an equilateral triangle as shown in the figure. They collide at the centroid $G$ of the triangle. After the collision, $A$ comes to rest, $B$ retraces its path with the speed $v=5{ \mathrm{m} \mathrm{s}}^{-1}.$ What is the speed of $C ?$

Figure shows a wedge $A$ of mass $6\mathrm{m}$ smooth semicircular groove of radius $a=8.4 \mathrm{m}$ placed on a smooth horizontal surface. $A$small block $B$ of mass $\mathrm{m}$ is released from a position in groove where its radius is horizontal. Find the speed (in ${\mathrm{ms}}^{-1}$ ) of bigger block when smaller block reaches its bottommost position.

A ball of mass $\mathrm{m}$ is allowed to roll down the wedge of mass $\mathrm{M}=2\mathrm{m}$ as shown in the figure. What is the displacement of wedge (in $\mathrm{m}$ ) when the ball reaches from $A$ to $B$ ? Take $\theta =45°, h=1 \mathrm{m}, d=4 \mathrm{m}$

A massless spring of force constant $1000 \mathrm{N} {\mathrm{m}}^{-1}$ is compressed a distance of $20 \mathrm{cm}$ between discs of $8 \mathrm{kg}$ and $2 \mathrm{kg}$, spring is not attached to discs. The system is given an initial velocity $3 \mathrm{m} {\mathrm{s}}^{-1}$ perpendicular to the length of spring as shown in the figure. What is the ground frame velocity of $2 \mathrm{kg}$ block in $(\mathrm{m} {\mathrm{s}}^{-1})$ when spring regains its natural length?

A child of mass $4 \mathrm{kg}$ jumps from cart $B$ to cart $A$ and then immediately back to cart $B.$ The mass of each cart is $20 \mathrm{kg}$ and they are initially at rest. In both the cases the child jumps at $6 \mathrm{m} {\mathrm{s}}^{-1}$ relative to the cart. If the cart moves along the same line with negligible friction with the final velocities of $v_B$ and $v_A,$ respectively, then the ratio of $6v_B$ and $5v_A$ is $x:1$. Write the value of $x$.