An object of mass $20 \,kg$ is displaced by $x = 5t^2 \,m$ (where $t$ is time) due to the application of a force. The ratio of the work done in $3 \,s$ and $5 \,s$ is:

$A$ block of mass $\sqrt{2} \, kg$ is released from the top of an inclined smooth surface as shown in the figure. If the spring constant of the spring is $100 \, N/m$ and the block comes to rest after compressing the spring by $1 \, m$,then the total distance travelled by the block along the incline before it comes to rest is ......... $m$.

$A$ box of mass $m$ is released from rest at position $1$ on the frictionless curved track shown. It travels a distance $d$ along the track in time $t$ to reach position $2$,falling a vertical distance $h$. Let $v$ be the instantaneous speed and $a$ be the instantaneous acceleration of the box at position $2$. Which of the following equations is valid for this situation?

$A$ car accelerates from rest to $u \ m/s$. The energy spent in this process is $E \ J$. The energy required to accelerate the car from $u \ m/s$ to $2u \ m/s$ is $nE \ J$. The value of $n$ is ..............

$A$ 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 $g = 10 \; ms^{-2}$)