In a smooth stationary cart of length $d$,a small block is projected along its length with velocity $v$ towards the front. The coefficient of restitution for each collision is $e$. The cart rests on a smooth ground and can move freely. The time taken by the block to come to rest with respect to the cart is:

$A$ block of mass $m = 0.1\,kg$ is connected to a spring of unknown spring constant $k.$ It is compressed to a distance $x$ from its equilibrium position and released from rest. After approaching half the distance $(\frac{x}{2})$ from the equilibrium position,it hits another block and comes to rest momentarily,while the other block moves with a velocity $3\,m/s.$ The total initial energy of the spring is ................ $J$.

$A$ ball strikes against the floor and returns with double the velocity; in which type of collision is it possible?

The sign of work done by a force on a body is important to understand. State carefully if the following quantities are positive or negative:
$(a)$ Work done by a man in lifting a bucket out of a well by means of a rope tied to the bucket.
$(b)$ Work done by gravitational force in the above case.
$(c)$ Work done by friction on a body sliding down an inclined plane.
$(d)$ Work done by an applied force on a body moving on a rough horizontal plane with uniform velocity.
$(e)$ Work done by the resistive force of air on a vibrating pendulum in bringing it to rest.

$A$ particle of mass $m$ moving horizontally with velocity $v_0$ strikes a smooth wedge of mass $M$,as shown in the figure. After the collision,the particle starts moving up the inclined face of the wedge and rises to a height $h$. The maximum height $h$ attained by the particle is:

The displacement $x$ of a particle moving in one dimension under the action of a constant force is related to the time $t$ by the equation $t = \sqrt{x} + 3$,where $x$ is in meters and $t$ is in seconds. The work done by the force in the first $6$ seconds is.....$J$