$A$ ball is dropped from a height of $5 \ m$ onto the floor of an elevator. The ball bounces off the floor. At the time of collision,the elevator is moving upwards with a velocity of $1 \ m/s$. What is the velocity of the ball immediately after the collision in $m/s$?

The graph below shows the variation of a force $F$ with time $t$ on a body which is moving in a straight line. The dependence of force on time is $F \propto t^{n}$. Initially,the body is at rest. If the speed of the object is $2 \, m/s$ at $3 \, s$,then the speed at $4 \, s$ will be approximately (in $m/s$):

$A$ 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]

In the figure,two blocks $M$ and $m$ are tied together with an inextensible and light string. The mass $M$ is placed on a rough horizontal surface with coefficient of friction $\mu$ and the mass $m$ is hanging vertically against a smooth vertical wall. The pulley is frictionless. Choose the correct statement$(s)$ related to the tension $T$ in the string.

$Assertion$: Two bodies of masses $M$ and $m$ $(M > m)$ are allowed to fall from the same height. If the air resistance for each is the same,then both bodies will reach the Earth simultaneously.
$Reason$: For the same air resistance,the acceleration of both bodies will be the same.

$A$ 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.