$A$ bullet of mass $25 \,g$ moving horizontally at a speed of $250 \,m/s$ is fired into a wooden block of mass $1 \,kg$ suspended by a long string. The bullet crosses the block and emerges on the other side. If the centre of mass of the block rises through a height of $20 \,cm$, find the speed of the bullet as it emerges from the block. (Take $g = 10 \,m/s^2$) (in $\,m/s$)

$A$ small disk can slide in a circular path on a frictionless inclined plane inclined at an angle of $30^\circ$ with the help of a thread as shown. The mass of the disk is $m$ and the acceleration due to gravity is $g$. If the disk is released when the thread is horizontal,the expression for the tension in the thread at the lowest point is: ................. $mg$

$A$ body is moved from rest along a straight line by a machine delivering constant power. The ratio of displacement and velocity $(s/v)$ varies with time $t$ as :

Three blocks $A, B$ and $C$ are kept as shown in the figure. The coefficient of friction between $A$ and $B$ is $0.2$,$B$ and $C$ is $0.1$,and $C$ and the ground is $0.0$. The masses of $A, B$ and $C$ are $3\, kg, 2\, kg$ and $1\, kg$ respectively. $A$ is given a horizontal velocity of $10\, m/s$. Blocks $A, B$ and $C$ always remain in contact and move together as a single system. The total work done by friction will be ........ $J$.

$A$ motor drives a body along a straight line with a constant force. The power $P$ developed by the motor must vary with time $t$ according to

$A$ $1.0\, kg$ block collides with a horizontal weightless spring of force constant $2.75\, N/m$ as shown in the figure. The block compresses the spring $4.0\, m$ from the rest position. If the coefficient of kinetic friction between the block and the horizontal surface is $0.25$,the speed of the block at the instant of collision is ................. $m/s$.