From 2^nd law of motion, F = ma

W = FS

or, W = ma(v²-u²)/2a [Since, v²-u² = 2aS So, S = (v²-u²)/2a]

or, W = m(v²-u²)/2 = ½ m(v²-u²)

If u = 0 then, W = ½ m(v²-0²) = ½ mv²

So, E_k = ½ mv²

Work done= Change in kinetic energy

or, W = E_kf -E_ki

or, W = ½ mv² – ½ mu²

Potential energy:-

The energy possessed by an object by virtue of its position or configuration is known as its potential energy. Ex:- Water stored in a dam.

Expression of potential energy:-

Consider an object having mass ‘m’ at a height ‘h’.

From 2^nd law of motion,

F = ma = mg (Where, g is acceleration due to gravity.)

Now, W = FS

or, W = mgh [here, S = h]

So, E_p= mgh

Law of conservation of energy:-

It states that energy can neither be created nor be destroyed. It can only be converted from one form to another.

Mechanical energy:-

The sum of kinetic energy & potential energy of an object is known as mechanical energy.

• Mechanical energy = Potential energy +Kinetic energy = constant

Mathematical proof of conservation of mechanical energy for a freely falling object:-

Consider an object having mass ‘m’ at height ‘h’.

At point A, velocity = 0, acceleration = g and height = h

K.E. = ½ mv² = ½ m.0² = 0

P.E. = mgh

M.E. = K.E.+P.E. = 0 + mgh = mgh

At point B,

Using, v²-u² = 2aS

or, v² = 0²+2gx [here,u=0, a=g & S=x]

or, v² = 2gx

K.E.= ½ mv² = ½ m.2gx = mgx

P.E.= mg(h-x) = mgh – mgx

So, M.E.=K.E.+P.E. = mgx + mgh – mgx = mgh

At point C,

Using, v²-u² = 2aS

or, v²= u²+2aS = 0²+2gh = 2gh

K.E.= ½ mv² = ½ m(2gh) = mgh

and P.E.= mgh = mg.0 = 0

so, M.E.= K.E.+P.E. = mgh+0 = mgh

Clearly, the total mechanical energy for a freely falling object is ‘mgh’ i.e., constant at every point.

Power:-

The rate of doing work is called power.

• Power= work/time (P=W/t)
• S.I. unit :- J/S or Watt (W)

1 watt of power:- The power of an agent is said to be 1 watt if it does 1 J work in 1 sec.

• 1 kW = 1000 W = 1000 J/s
• Commercial unit of energy :- kilowatt hour (kW h)

Conversion of commercial unit of energy to S.I. unit of energy:-

1 kwh = 1000 W x 3600 sec = 3.6 x10⁶ W sec = 3.6 x10⁶ J

or, 1 unit = 1 kwh = 3.6 x 10⁶ J