$A$ block $C$ of mass $m$ is moving with velocity $v_0$ and collides elastically with block $A$ of mass $m$ which is connected to another block $B$ of mass $2m$ through a spring of spring constant $k$. What is $k$ if $x_0$ is the compression of the spring when the velocity of $A$ and $B$ is the same?

An athlete throws a shotput of mass $25 \,kg$ with an initial speed of $4 \,ms^{-1}$ at an angle of $45^{\circ}$ with the horizontal from a height of $2 \,m$ above the ground. Assuming air resistance to be negligible,the kinetic energy of the shotput when it just touches the ground is . . . . . . $\left(g=10 \,ms^{-2}\right)$ (in $\,J$)

$A$ man of mass $60 \ kg$ gets $10^5 \ calories$ of energy from food. If his efficiency is $28\%$,up to what height in $m$ can he climb?

$A$ person trying to lose weight (dieter) lifts a $10\; kg$ mass,one thousand times,to a height of $0.5\; m$ each time. Assume that the potential energy lost each time she lowers the mass is dissipated.
$(a)$ How much work does she do against the gravitational force?
$(b)$ Fat supplies $3.8 \times 10^{7} \;J$ of energy per kilogram which is converted to mechanical energy with a $20\%$ efficiency rate. How much fat will the dieter use up?

$A$ body of mass $4 \,kg$ is moving with momentum of $8 \,kg \,m/s$. $A$ force of $0.2 \,N$ acts on it in the direction of motion of the body for $10 \,s$. The increase in kinetic energy in joules is

$A$ curved surface is shown in the figure. The portion $BCD$ is free of friction. There are three spherical balls of identical radii and masses. Balls are released from rest one by one from $A$,which is at a slightly greater height than $C$.
With the surface $AB$,ball $1$ has large enough friction to cause rolling down without slipping; ball $2$ has a small friction and ball $3$ has a negligible friction.
$(a)$ For which balls is total mechanical energy conserved?
$(b)$ Which ball$(s)$ can reach $D$?
$(c)$ For balls which do not reach $D$,which of the balls can reach back $A$?