Question

Proof law of conservation of energy when a ball is projects upward

Answer 1

Let us strat from point A as shown in figure. Let us assume initial energy is given to the body so that it moves vertically with speed V.

 

initial energy E = (1/2)mV2...........................(1)

 

Let us consider a point at B which is at a height h above the ground level. Let v be the speed of body at B. Total energy of the body is given by

 

Energy at B = potential energy + kinetic energy = mgh+(1/2)mv2  ...............(2)

 

But as per equation of motion v2 = V2-2gh    ............(3)

 

substituting for v2 using (3) in (2), we get Energy at B = mgh+(1/2)m[V2-2gh] = (1/2)mV2 .............................(4)

 

Let C be the highest point the body can reach. Since the body comes to zero speed at this point, we have only potential energy

 

Energy at C = mgH ................................(5)

 

As per equation of motion we have, 0 = V 2-2gH ;  otherwise H = V 2/(2g) ..............(6)

 

By substituting for H using (6) in (5),  we get Energy at C = mgH = (1/2)mV2 ...............................(7)

 

Equation (7) shows that initial energy is fully converted into potential energy at the maximum height and both are same.

 

Let us assume during downward movement, at B, the falling body has the speed u m/s. Total energy EB at B during downward movement is given by

 

EB  = potential energy+kinetic energy = mgh+(1/2)mu2 .................(8)

 

as per equation of motion, u2 = 0+2g(H-h) ...................(9)

 

By substituting for u2 from (9) in (8), we habe EB = mgh+(1/2)m{ 2g(H-h) } = mgH .....................(10)

 

Finally when the body reaches A, it has only kinetic energy and its potential energy is zero

 

Energy EA at A at end of the downward movement is given by

 

EA = (1/2)mU2 ..................(11)

 

As per equation of motion U2 = 0+2gH ...............(12)

 

By substituting for U2 from (12) in (11), we get EA = (1/2)m{2gH} = (1/2)mgH ...................(13)

 

From equations (1), (4), (7), (10) and (13), we know total energy of the body is always constant. This proves the law of conservation of energy.......

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