Energy forms: kinetic and potential

ENERGY:

Energy is defined as the ability to do work. The S.I unit of energy is joule (J).

KINETIC ENERGY:

Energy possessed by an object due to its motion is called kinetic energy.

or

The work required to accelerate a body of a given mass from rest to its stated velocity.

The S.I unit of kinetic energy is joule.

As we know that kinetic energy is due to the motion of object. Therefore for an object of mass $m$ moving with speed $v$ kinetic energy depends upon:

• The mass $m$ of the object- the greater the mass, the greater its K.E

• The speed $v$ of the object- the greater the speed, the greater the K.E

Mathematically:

$E_k = \frac {1}{2} \; mv^2$

Examples

• A car moving at high speed will possess Kinetic energy

• Moving particles in a gas have Kinetic Energy

DERIVATION OF KINETIC ENERGY:

Consider a body of mass $m$ initially moving with the velocity $v_i=v$ . When an opposing force $F$ is applied on body, it eventually comes to rest $v_f=0$ after covering the displacement $s$. The change in kinetic energy is equal to the work done on the body, therefore:

$\Delta E_k = W$

$W = F \times \Delta s$

where, $s =$ displacement

From Newton’s First Law we know $F= m\times a$

The equation becomes:

$\Delta E_k = W = m\times a\times\Delta s$ ……… equ (i)

From the 3rd equation of motion we know that:

$2a\Delta s = {v_f}^2 - {v_i}^2$

So,

$a\Delta s = \frac {{v_f}^2 - {v_i}^2} {2}$ ……… equ (ii)

substituting equation (ii) into equation (i):

$\Delta E_k = m \times$ $\frac {{v_f}^2 - {v_i}^2} {2}$

$\Delta E_k = \frac {1}{2} \times (\;m {v_f}^2 - m {v_i}^2)$

Now we already know that kinetic energy is the energy that it possessed due to its motion. So the kinetic energy at rest should be zero.

$\Delta E_k = \frac {1}{2} \times (\;m ({0})^2 - m {v}^2)$ ;

$v_i=v$

Therefore, we can say that kinetic energy is:

$\boxed{E_k = \frac {1}{2} \; mv^2}$

POTENTIAL ENERGY:

Potential energy of a body is defined as the energy that a body possesses by virtue of its position, shape or state of a system.

S.I. unit of potential energy is Joule (J).

There are different types of potential energy.

• Gravitational potential energy

• Elastic potential energy

• Chemical potential energy

Examples:

• A body raised to a height “h” above the ground has gravitational potential energy.

• A stretched spring has elastic potential energy due to its stretched position (condition).

• The energy stored in the plants that we eat is chemical potential energy.

GRAVITATIONAL POTENTIAL ENERGY:

It is the energy stored in an object due to its height.

or

It is also defined as the work done stored in a body in lifting it to a height “h”.

Mathematically:

$E_p = m\, g\,h$

$m=$ mass of the object

$g=$ acceleration due to gravity

$h=$ height at which the object is stored

$E_p =$ Gravitational Potential Energy

DERIVATION OF GRAVITATAIONAL POTENTIAL ENERGY:

To derive the expression for gravitational potential energy, let us consider an object of mass “$m$” which is raised up through height “$h$” from the ground:

The work done in lifting it to height “h” is stored in it as its gravitational potential energy ($E_p$), i.e.

$E_p$ = Work done ($W$) $E_p = F \times d$

The distance covered is vertical, i.e. the height ($h$) of the object.

$E_p = F \times h$

From newton’s 2nd law we know:

$F = m\times a$

Since, it is work done against gravity, we can use the acceleration due to gravity ($g$) here.

Therefore equation becomes:

$\boxed{E_p = m\times g \times h}$