$A$ bead of mass $\frac{1}{2} \, kg$ starts from rest from point $A$ to move in a vertical plane along a smooth fixed quarter ring of radius $5 \, m$,under the action of a constant horizontal force $F = 5 \, N$ as shown. The speed of the bead as it reaches point $B$ is ................ $m/s$. [Take $g = 10 \, m/s^2$]

$A$ particle $(m = 1 \; kg)$ slides down a frictionless track $(AOC)$ starting from rest at a point $A$ (height $2 \; m$). After reaching $C$,the particle continues to move freely in air as a projectile. When it reaches its highest point $P$ (height $1 \; m$),the kinetic energy of the particle (in $J$) is: (Figure drawn is schematic and not to scale; take $g = 10 \; ms^{-2}$)

An object of mass $8 \ kg$ moves under the influence of a force. The position of the object varies with time as $x = \frac{1}{2} t^2$,where $x$ is in meters and $t$ is in seconds. Find the work done by the force in the first $2 \ s$ in $J$.

The position $x$ of a particle moving in one dimension under the influence of a constant force is given by $t = \sqrt{x} + 3$, where $x$ is in meters and $t$ is in seconds. Find the work done in the first $6$ seconds. (in $\text{ J}$)

$A$ mass $m$ slips along the wall of a semispherical surface of radius $R$. The velocity at the bottom of the surface is

$A$ particle of mass $m$ is moving along a circle of radius $R$ such that its tangential acceleration $a_t$ varies with distance covered $x$ as $a_t = \alpha x^2$, where $\alpha$ is a constant. The kinetic energy $K$ of the particle varies with the distance as $K = \beta x^c$, where $\beta$ and $c$ are constants. The values of $\beta$ and $c$ are: