$A$ block of mass $1 \ kg$,moving along the $x$-axis with an initial speed $v_{i} = 10 \ m/s$,enters a rough region ranging from $x = 0.1 \ m$ to $x = 1.9 \ m$. The retarding force acting on the block in this range is $F_{r} = -kx \ N$,where $k = 10 \ N/m$. Find the final speed of the block as it crosses the rough region.

The figure shows the frictional force versus displacement for a particle in motion. The loss of kinetic energy in travelling over $s = 0$ to $20\, m$ will be......$J$.

$A$ 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

$A$ position-dependent force $F = 7 - 2x + 3x^2 \, N$ acts on a small body of mass $2 \, kg$ and displaces it from $x = 0$ to $x = 5 \, m$. The work done in joules is

$A$ force $(3x^2 + 2x - 5) \text{ N}$ displaces a body from $x = 2 \text{ m}$ to $x = 4 \text{ m}$. The work done by this force is .......... $J$.

$A$ block of mass $2\,kg$ moving on a horizontal surface with speed of $4\,ms^{-1}$ enters a rough surface ranging from $x = 0.5\,m$ to $x = 1.5\,m$. The retarding force in this range of rough surface is related to distance by $F = -kx$ where $k = 12\,Nm^{-1}$. The speed of the block as it just crosses the rough surface will be ........... $ms^{-1}$.