Work Energy And Power

Kinetic energy is related to momentum through the relation

or
Given, ,


Increase in kinetic energy

Hence, percentage increase in kinetic energy

In an elastic collision, linear momentum and kinetic energy are conserved.
Initial momentum Final momentum
i.e.,

or
or
Again kinetic energy is also conserved.


or
or
Dividing E(ii) by E (i), we get

or As
so

Note: The conservation of momentum and the conservation of total energy holds for all the three types of collisions, but KE conservation law holds only for perfectly elastic collision.

Key Idea: Equate the momenta of the system along two perpendicular axes.
Let be the velocity and the direction of the third piece as shown.

Equating the momenta of the system along OA and OB to zero, we get
... (i)
and ... (ii)
These give
or

Thus,
Putting the value of in E (i), we get

The third piece will move with a velocity of in a direction making an angle of with either piece.
Alternative: Key Idea: The square of momentum of third piece is equal to sum of squares of momentum of first and second piece. As from key idea,

or
or
or

Key Idea: Potential energy of block at height h is converted into potential energy of spring. When block is at height h, it has potential energy which is converted into potential energy of spring to conserve energy i.e., where k is force constant of spring.

Potential energy is converted into kinetic energy of rotation.
From law of conservation of energy, we have the potential energy of rod when it is vertical is converted to kinetic energy of rotation.

Moment of inertia of a thin rod is


Given,

Also,

Power is the rate at which a force does work.
Efficiency of engine,
Thus, power
Given, and
Hence, power