Which of the following statements is incorrect?

$A$ student skates up a ramp that makes an angle $30^{\circ}$ with the horizontal. He/she starts (as shown in the figure) at the bottom of the ramp with speed $v_0$ and wants to turn around over a semicircular path $xyz$ of radius $R$ during which he/she reaches a maximum height $h$ (at point $y$) from the ground as shown in the figure. Assume that the energy loss is negligible and the force required for this turn at the highest point is provided by his/her weight only. Then ($g$ is the acceleration due to gravity):
$(A)$ $v_0^2 - 2gh = \frac{1}{2} gR$
$(B)$ $v_0^2 - 2gh = \frac{\sqrt{3}}{2} gR$
$(C)$ The centripetal force required at points $x$ and $z$ is zero.
$(D)$ The centripetal force required is maximum at points $x$ and $z$.

State whether the following statements are true or false:
$(a)$ If the magnitude of force and length are increased by $4$ times,the magnitude of energy increases by $16$ times.
$(b)$ In an inelastic collision,both momentum and energy are conserved.
$(c)$ If work is done on a system by non-conservative forces,the potential energy increases.

$A$ tennis ball dropped from a height of $2 \,m$ rebounds only $1.5 \,m$ after hitting the ground. What fraction of its energy is lost in the impact?

$A$ shell at rest on a smooth horizontal surface explodes into two fragments of masses $m_1$ and $m_2$. If just after explosion $m_1$ moves with speed $u$,then the work done by internal forces during the explosion is:

$A$ block of mass $0.18 \ kg$ is attached to a spring of force constant $2 \ N/m$. The coefficient of friction between the block and the floor is $0.1$. Initially,the block is at rest and the spring is unstretched. The block is pushed as shown in the figure. The block slides a distance of $0.06 \ m$ and comes to rest. If the initial velocity of the block is $V = N/10 \ m/s$,then what is the value of $N$?