The track $OABCD$ (as shown in figure) is smooth. What minimum speed has to be given to a particle lying at point $A$ so that it can reach point $C$? ( Take, $g=10 \mathrm{m} {\mathrm{s}}^{-2}$.)

A toy car of mass $5 \mathrm{kg}$ moves up a ramp under the influence of force $F$ plotted against displacement $x$. The maximum height attained by the car is given by,

A $2 \mathrm{kg}$ block is dropped from a height $0.4 \mathrm{m}$ on a spring of force constant, $k=1960 \mathrm{N} {\mathrm{m}}^{-1}$. The maximum compression of the spring is (spring is fixed at it's lower end rigidly),

In the figure shown, find the horizontal velocity of the block when it reaches point $Q.$ (Assume the surface to be frictionless and take $g=9.8 \mathrm{m} {\mathrm{s}}^{-2}$.)

A block of mass, $m=2 \mathrm{kg}$ is attached to two unstretched springs of force constant, $k_1=100 \mathrm{N} {\mathrm{m}}^{-1}$ and $k_2=125 \mathrm{N} {\mathrm{m}}^{-1}$. The block is displaced towards the left through a distance of $10 \mathrm{cm}$ and released. Find the speed of the block as it passes through the mean position.

A girl slides along a curved water slide, without friction, from a height $h$. What is the maximum value of height $y$ in terms of $h$ and $\mathrm{\theta }$?