A body of mass $10\, kg$ is released from a tower of height $20\,m$ and body acquires a velocity of $10\,ms^{-1}$ after falling through the distance $20\,m$ . The work done by the push of the air on the body is:- ................. $\mathrm{J}$  (Take $g = 10\, m/s^2$ )

A block of mass $m=1\; kg$. moving on a horizontal surface with speed $v_{t}=2 \;m s ^{-1}$ enters a rough patch ranging from $x=0.10\; m$ to $x=2.01\; m .$ The retarding force $F$ on the block in this range is inversely proportional to $x$ over this range,

$F_{r}=\frac{-k}{x}$ for $0.1 < x < 2.01 \;m$

$=0$ for $x < 0.1\; m$ and $x > 2.01\; m$

where $k=0.5\; J .$ What is the final kinetic energy and speed $v_{f}$ of the block as it crosses this patch ?

A rocket accelerates straight up by ejecting gas downwards. In a small time interval $\Delta t$, it ejects a gas of mass $\Delta m$ at a relative speed $u$. Calculate $KE$ of the entire system at $t + \Delta t$ and $t$ and show that the device that ejects gas does work $=(\frac {1}{2})\Delta mu^2$ in this time interval (negative gravity).

A bullet looses ${\left( {\frac{1}{n}} \right)^{th}}$ of its velocity passing through one plank. The number of such planks that are required to stop the bullet can be

A particle of mass $m$ slides from rest down a plane inclined at $30^o$ to the horizontal. The force of resistance acting on the particle during motion is $ms^2$ where $s$ is the displacement of the particle from its initial position. The velocity (in $m/s$) of the particle when $s = 1\,m$ is $v$. The value of $\frac{3v^2}{14}$ is :-

A bullet of mass $10\, g$ moving horizontally with a velocity of $400\, m s^{-1}$ strikes a wood block of mass $2\, kg$ which is suspended by light inextensible string of length $5\, m.$ As a result, the centre of gravity of the block found to rise a vertical distance of $10\, cm.$ The speed of the bullet after it emerges out horizontally from the block will be ................... $\mathrm{ms}^{-1}$