$A$ block moving horizontally on a smooth surface with a speed of $40 \, m/s$ splits into two equal parts. If one of the parts moves at $60 \, m/s$ in the same direction,then the fractional change in the kinetic energy will be $x: 4$ where $x = ..... .$

Identify the correct statements from the following:
$(A)$ Work done by a man in lifting a bucket out of a well by means of a rope tied to the bucket is negative.
$(B)$ Work done by gravitational force in lifting a bucket out of a well by a rope tied to the bucket is negative.
$(C)$ Work done by friction on a body sliding down an inclined plane is positive.
$(D)$ Work done by an applied force on a body moving on a rough horizontal plane with uniform velocity is zero.
$(E)$ Work done by the air resistance on an oscillating pendulum is negative.
Choose the correct answer from the options given below:

$A$ particle of mass $m$ moving with velocity $V_0$ strikes a simple pendulum of mass $m$ and sticks to it. The maximum height attained by the pendulum will be

$A$ man throws a ball with a speed of $12 \ m/s$ at a height of $12 \ m$. If he throws the ball such that it just reaches this height,what is the percentage of energy saved?

$A$ bullet of mass $10 \,g$ is fired horizontally with a velocity $1000 \,ms^{-1}$ from a rifle situated at a height $50 \,m$ above the ground. If the bullet reaches the ground with a velocity $500 \,ms^{-1}$, the work done against air resistance in the trajectory of the bullet is : $(g=10 \,ms^{-2})$ (in $\,J$)

Two bodies $A$ and $B$ of mass $m$ and $2m$ respectively are placed on a smooth floor. They are connected by a spring of negligible mass. $A$ third body $C$ of mass $m$ is placed on the floor. The body $C$ moves with a velocity $v_0$ along the line joining $A$ and $B$ and collides elastically with $A$. At a certain time after the collision,it is found that the instantaneous velocities of $A$ and $B$ are the same and the compression of the spring is $x_0$. The spring constant $k$ will be