Mathematically  prove that the total mechanical energy of a freely falling body is constant at all positions .

Asked by Taruna | 10th Sep, 2015, 06:44: PM

Expert Answer:

Let us consider an object of mass, ‘m’ is thrown upwards to a height ‘h’ above the ground level.

At point A:

The body must be at rest and v = 0

PEA = mgh and KEA = 0 

Total energy = PEA +  KEA = mgh ......... (Equation 1)

When the body drops down, let us say it covers a distance ‘x’ at point C. Thus, its height from the ground is ‘h – x’.

At point C:

PEC = mg(h-x) = mgh – mgx  ........ (Equation 2)

begin mathsize 12px style KE subscript straight C space equals space 1 half mv squared space........ space left parenthesis Equation space 3 right parenthesis Using space third space equation space of space motion comma space we space get straight v squared space equals space straight u squared space plus space 2 as Here comma space straight u space equals space 0 straight v squared space equals space 0 space plus space 2 ax straight v squared space equals space 2 gx space space........ space left parenthesis Equation space 4 right parenthesis Substituting space the space above space equation space in space equation space left parenthesis 3 right parenthesis comma space we space get KE subscript straight C space equals space 1 half straight m open parentheses 2 gx close parentheses space equals space mgx Total space energy space equals space PE subscript straight C space plus space KE subscript straight C space space space space space space space space space space space space space space space space space space space equals space mgh space – space mgx space plus space mgx space Total space energy space equals space mgh space........ space left parenthesis Equation space 5 right parenthesis end style
At point B:

The body hits the ground u = 0 and h = 0

So, it possesses only kinetic energy.

PEA = 0 

begin mathsize 12px style KE subscript straight B space equals space 1 half mv squared space........ space left parenthesis Eqaution space 6 right parenthesis straight v squared space equals space straight u squared space plus space 2 as straight v squared space equals space 2 gh Substituting space the space above space in space equation space 6 comma space we space get KE subscript straight B space equals space 1 half straight m 2 gh space equals space mgh Total space energy space equals space PE subscript straight B space plus space KE subscript straight B space space space space space space space space space space space space space space space space space space space equals space mgh space space........ space left parenthesis Eqaution space 7 right parenthesis end style

Thus, from equation (1), (5) and (7) that the total energy remains constant.

Answered by Yashvanti Jain | 11th Sep, 2015, 10:25: AM