The graph of the resistive force acting on an object versus the distance covered by it is shown in the figure. The mass of the object is $2 \ kg$ and its initial velocity is $10 \ m/s$. When the distance covered by the object is $4 \ m$,its kinetic energy will be $.... \ J$. (in $J$)

Power applied to a particle varies with time as $P = (4t^3 - 5t + 2) \, W$,where $t$ is in seconds. Find the change in its $K.E.$ between time $t = 2 \, s$ and $t = 4 \, s$. ............... $J$

The work done by a force $\vec{F} = (-6x^3 \hat{i}) \, N$ in displacing a particle from $x = 4 \, m$ to $x = -2 \, m$ is ............... $J$.

$A$ particle of mass $0.1 \, kg$ is subjected to a force which varies with distance as shown in the figure. If it starts its journey from rest at $x = 0$,its velocity at $x = 12 \, m$ is .......... $m/s$.

The work done on a particle of mass $m$ by a force,$F = K \left[ \frac{x}{(x^2+y^2)^{3/2}} \hat{i} + \frac{y}{(x^2+y^2)^{3/2}} \hat{j} \right]$ (where $K$ is a constant of appropriate dimensions),when the particle is moved from the point $(a, 0)$ to the point $(0, a)$ along a circular path of radius $a$ about the origin in the $x-y$ plane is:

$A$ bullet enters a piece of wood with velocity $v_0$ and the resistive force acting on the bullet in the wood is proportional to $v^{\frac{1}{3}}$. If the total distance travelled by the bullet is proportional to $(v_0)^\beta$,then the value of $\beta$ is