A particle moves in a straight line with retardation proportional to its displacement. Its loss of kinetic energy for any displacement $x$  is proportional to

A body of mass $0.5\, kg$ travels on straight line path with velocity $v =\left(3 x ^{2}+4\right) m / s$. The net workdone by the force during its displacement from $x =0$ to $x =2\, m$ is$.......J$

A lorry and a car moving with the same $K.E.$ are brought to rest by applying the same retarding force, then

A body of $0.5 \,kg$ moves along the positive $X$-axis under the influence of a varying force $F$ (in newton) as shown below.If the speed of the object at $x=4 \;m$ is $3.16 \,ms ^{-1}$, then its speed at $x=8 \,m$ is ................. $\,ms ^{-1}$

A particle of mass $10\,g$ moves in a straight line with retarcation $2x$, where $x$ is the displacement in $SI$ units. Its loss of kinetic energy for above displacement is $\left(\frac{10}{x}\right)^{- n }\, J$. The value of $n$ will be $............$.

A particle of mass $500 \,gm$ is moving in a straight line with velocity $v=b x^{5 / 2}$. The work done by the net force during its displacement from $x=0$ to $x =4 \,m$ is ...................$J$ (Take $b =0.25 \,m ^{-3 / 2} s ^{-1}$ )