$A$ ball of mass $m$ is released from the top of a smooth movable wedge of mass $m$. When the ball collides with the floor,the velocity of the wedge is $v$. Then the maximum height attained by the ball after an elastic collision with the floor is: (Neglect any edge at the lower end of the wedge).

$A$ disc of mass $0.2 \ kg$ is kept floating in air without falling by vertically firing bullets each of mass $0.05 \ kg$ on the disc at the rate of $10$ bullets per second. If the bullets rebound with the same speed,then the speed of each bullet is (Acceleration due to gravity $g = 10 \ m \ s^{-2}$) (in $m \ s^{-1}$)

Write important guidance for solving problems in mechanics.

$A$ block of mass $2M$ is attached to a massless spring with spring-constant $k$. This block is connected to two other blocks of masses $M$ and $2M$ using two massless pulleys and strings. The accelerations of the blocks are $a_1, a_2$ and $a_3$ as shown in the figure. The system is released from rest with the spring in its unstretched state. The maximum extension of the spring is $x_0$. Which of the following option$(s)$ is/are correct? [$g$ is the acceleration due to gravity. Neglect friction]

$A$ helicopter of mass $2000 \,kg$ rises with a vertical acceleration of $15 \,m s^{-2}$. The total mass of the crew and passengers is $500 \,kg$. Calculate the magnitude and direction of the following (take $g = 10 \,m s^{-2}$):
$(a)$ Force on the floor of the helicopter by the crew and passengers.
$(b)$ Action of the rotor of the helicopter on the surrounding air.
$(c)$ Force on the helicopter due to the surrounding air.

$A$ $1 \,kg$ block situated on a rough incline is connected to a spring of negligible mass having spring constant $100 \,N \,m^{-1}$ as shown in the figure. The block is released from rest with the spring in the unstretched position. The block moves $10 \,cm$ down the incline before coming to rest. The coefficient of friction between the block and the incline is (Take $g=10 \,m \,s^{-2}$ and assume that the pulley is frictionless).