Revision: Class 11 >> Mechanical Energy Dynamics, Power, and Collisions NEET (UG) Mechanical Energy Dynamics, Power, and Collisions

The scalar product or dot product of two nonzero vectors \[\vec P\] and \[\vec Q\] is defined as the product of the magnitudes of the two vectors and the cosine of the angle θ between the two vectors.

The motion of a body in a straight line path is called translational motion.

When a force acts on a rigid body which is free to move, the body starts moving in a straight line in the direction of the force. This is called translational motion.

“Capacity of doing work” is called Energy.

Energy, in physics, is the capacity for doing work. It may exist in potential, kinetic, thermal, electrical, chemical, nuclear, or other various forms. There are, moreover, heat and work-i.e., energy in the process of transfer from one body to another.

The work done by a force on a body is equal to the product of the force applied and the distance moved by the body in the direction of force i.e.,

Work done = Force × distance moved in the direction of force

The SI unit of work is joule.
1 joule of work is said to be done when a force of 1 newton displaces a body through 1 metre in its own direction.

The kinetic energy of the body due to motion in a straight line is called translational kinetic energy.

If a body moves to and fro about its mean position, the motion is called vibrational motion.

The kinetic energy of the body due to rotational motion is called rotational kinetic energy or simply rotational energy.

If a body rotates about an axis, the motion is called rotational motion.

The energy possessed by a body due to its state of motion is called its kinetic energy.

Work is said to be done only when the force applied on a body makes the body move (i.e., there is a displacement of the body). 

The kinetic energy of the body due to its vibrational motion is called vibrational kinetic energy or simply vibrational energy.

"If there is a decrease in the potential energy (like a body falling down) due to a conservative force, it is entirely converted into kinetic energy. Work done by the force then appears as kinetic energy. Vice versa if an object is moving against a conservative force its kinetic energy decreases by an amount equal to the work done against the force."

The energy possessed by a body by virtue of its specific position (or changed configuration) is called the potential energy.

The energy possessed by a body due to its state of rest or of motion, is called mechanical energy.

The energy possessed by a body at rest due to its position or size and shape is called potential energy.

The rate of doing work is called power.

Power is defined as the rate of doing work or work done per second.

i.e., Power = `("Work done in joule")/("Times in second")`

or,  p = `("W (in joule)")/("t (in second)")`

The source describes a collision as a process where "several objects come together, interact (exert forces on each other) and scatter in different directions."

For two colliding bodies, the negative of the ratio of the relative velocity of separation to the relative velocity of approach is called the coefficient of restitution.

An event where two or more objects come into contact and exert forces on each other, causing changes in their individual momenta while the total momentum of the system remains conserved.

∑KE_initial = ∑KE_final

Where:

• ∑KE_initial = total initial kinetic energy
• ∑KE_final = total final kinetic energy

A collision where there is a loss in kinetic energy during the collision, but linear momentum is conserved.

A collision where the kinetic energy of the entire system is conserved during the collision (along with the linear momentum).

A collision in two dimensions (oblique collision) is defined as: a collision where the direction of at least one initial velocity is NOT along the line of impact, requiring analysis using two mutually perpendicular directions—the line of impact and the common tangent at the point of contact.

Oblique or non-head-on collision.

A force is said to be a conservative force if the work done by or against it is independent of the actual path chosen and depends only on the initial and final positions of the object.

A force that does not follow the conservative force rule, where the work done by or against it depends on the actual path taken.

K = \[\frac {1}{2}\] mv²

Kinetic Energy = \[\frac {1}{2}\] mass × (velocity)²

Gravitational Potential Energy U_h = mgh

Power P = \[\frac{\text{Work done }W}{\text{Time taken }t}\]

or

P = \[\frac {W}{t}\]

\[\sum\vec{p}_{initial}\] = \[\sum\vec{p}_{final}\]

Where:

• \[\sum\vec{p}_{initial}\] = total initial momentum of all objects
• \[\sum\vec{p}_{final}\] = total final momentum of all objects

∑KE_initial > ∑KE_final

Where:

• The lost kinetic energy is due to internal friction or vibrational motion of atoms, causing a heating effect
• Linear momentum is still conserved: \[\sum\vec{p}_{initial}\] = \[\sum\vec{p}_{final}\]

Statement:

According to the work-energy theorem, the increase in kinetic energy of a moving body is equal to the work done by a force acting in the direction of the moving body.

Proof:

Let a body of mass m be moving with an initial velocity u. When a constant force F is applied to the body along its direction of motion, it produces an acceleration a, and the body's velocity increases from u to v over a distance S.

Force,

F = ma

Work done by the force,

W = F × S

From the equation of motion,

\[v^2=u^2+2aS\Rightarrow S=\frac{v^2-u^2}{2a}\]

Substituting equations (i) and (iii) into (ii):

W = \[ma\times\frac{v^2-u^2}{2a}=\frac{1}{2}m(v^2-u^2)\]

Now,
Initial kinetic energy, K_i = \[\frac {1}{2}\]mu²
Final kinetic energy, K_f = \[\frac {1}{2}\]mv²

Therefore,

W = K_f − K_i

Conclusion:

Work done on the body = Increase in its kinetic energy.
Hence, the work-energy theorem is proved.

• The S.I. unit of work is joule (J).
    1 joule = 1 newton × 1 metre, i.e., work done when a force of 1 N moves a body 1 m in its direction.
• The C.G.S. unit of work is erg.
    1 erg = 1 dyne × 1 cm, i.e., work done when a force of 1 dyne moves a body 1 cm in its direction.
• The relation between joule and erg is:
    1 joule = 10⁷ erg

• There are two main types of potential energy: gravitational and elastic.
• Gravitational potential energy is due to height and is given by U = mgh.
• It is zero at infinity and becomes less negative as the distance from Earth increases.
• Elastic potential energy is stored when an object is stretched or compressed.
• Lifting a body stores energy as gravitational potential energy by doing work against gravity.

• S.I. unit: If 1 joule of work is done in 1 second, the power spent is said to be 1 watt.
• C.G.S. unit: The C.G.S. unit of power is erg per second (erg s^-1).
• Relationship between S.I. and C.G.S. units:
   1 W = 1 J s^-1 = 10⁷ erg s^-1
• 1 horse power (H.P.) = 746 W = 0.746 kW