Two particles,of masses $M$ and $2M$,moving as shown,with speeds of $10\, m/s$ and $5\, m/s$,collide elastically at the origin. After the collision,they move along the indicated directions with speeds $v_1$ and $v_2$ respectively. The values of $v_1$ and $v_2$ are nearly:

Two bodies of different masses $m_1$ and $m_2$ have equal momenta. Their kinetic energies $E_1$ and $E_2$ are in the ratio:

$A$ target is made of two plates,one of wood and the other of iron. The thickness of the wooden plate is $4\,cm$ and that of the iron plate is $2\,cm$. $A$ bullet fired goes through the wood first and then penetrates $1\,cm$ into the iron. $A$ similar bullet fired with the same velocity from the opposite direction goes through the iron first and then penetrates $2\,cm$ into the wood. If $a_1$ and $a_2$ are the retardations offered to the bullet by the wood and iron plates respectively,then:

$A$ ball of mass $m_1$ falls from height $h_1$ from rest to strike a spring of force constant $K$,which forces another ball of mass $m_2$ to jump on a horizontal floor at a height $h_2$ below it. Find the horizontal distance at which the ball of mass $m_2$ strikes the floor from the position of the start. [Assume the spring is fixed and the collision is elastic].

$A$ ball is projected vertically upwards. Air resistance and variation in $g$ may be neglected. The ball rises to its maximum height $H$ in a time $T$,the height being $h$ after a time $t$.
[$1$] The graph of kinetic energy $E_k$ of the ball against height $h$ is shown in figure $1$.
[$2$] The graph of height $h$ against time $t$ is shown in figure $2$.
[$3$] The graph of gravitational potential energy $E_g$ of the ball against height $h$ is shown in figure $3$.
Which of the following shows the correct answers?

In the diagram shown,there is no friction at any contact surface. Initially,the spring has no deformation. What will be the maximum deformation in the spring? Consider all the strings to be sufficiently large. Consider the spring constant to be $K$.