The initial kinetic energy of the system is given by :
or
Case (i):- The final kinetic energy is :
Thus the kinetic energy is not conserved in this case.
Case (ii):- The final kinetic energy is :
...

The momentum is conserved in the collision as no external force is acting on the system. In the given case the rebound velocity is the same as the initial velocity thus the kinetic energy of the molecule initially and finally are same. Hence this is an elastic collision.

The volume of the drop is :
Thus the mass of raindrop is :
or
Thus the work done is given by :
or
or ...

The kinetic energy of the electron is given by :
or
Thus velocity is obtained as :
or
Similarly, we can find the velocity of the proton :
...

We know that the power is given by :
or
or
It is given that power is constant, thus :
or
By integrating both sides, we get
...

It is given that acceleration is constant thus force will also be constant (by Newton's law of motion F = ma).
Also,
or
Thus
Now, the work done by the force is given by :
Hence power is directly...

Since the potential energy of the system depends upon the separation between the bodies thus the forces acting on the body are conservative in nature. We know that conservative forces produce elastic collisions.

The total kinetic energy of the system cannot be conserved in case of inelastic collision as there is loss of energy in the form of deformation. But the total linear momentum of the system remains constant even in the case of inelastic collision as no external force is acting.

Yes, in case of elastic collision the total linear momentum of the system remains conserved as no external force is acting on the system of balls.

No, because at the time of the collision, the kinetic energy is converted to the potential energy. Thus total kinetic energy is not constant at the collision.

True but not always:- In the case of inelastic collisions, few amounts of energy is converted into other forms of energy such as sound or in deformation. Thus final kinetic energy is always less as compared to initial kinetic energy. But in case of the explosion of a bomb final kinetic energy is greater than the initial kinetic energy

False:- This is true only for conservative forces e.g. gravitational force. For e.g in case of frictional force (non-conservative force), the work done in a closed-loop cannot be zero as energy is wasted throughout.

False:- Internal forces will not change the energy of the system but external forces can change the total energy by changing their magnitude or direction.

False:- The linear momentum and energy will be conserved if both are considered in a system. But for individual bodies, this conservation of momentum and energy doesn't hold. This is because the impact during the collision may transfer energy/momentum of one ball to the other ball.

The conservation of total linear momentum doesn't depend upon the fact whether it is an elastic collision or an inelastic collision.

The internal force cannot produce a change in the total momentum as no external force is acting. Thus the change in total momentum is proportional to the external forces acting on the body.

Work done by the body against friction results in a decrease in the velocity of the body. Thus the kinetic energy of the body decreases.

It is given that work done by the conservative force is positive, thus the force acts in the direction of the motion. This results in a decrease in distance between the bodies. Thus it's potential energy decreases.

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