Question:
a body is thrown vertically up with certain initial velocity. the potential and kinetic energies of the body are equal at a point p in its path. if the same body is thrown with double the velocity upwards the ratio of potential and kinetic energies of the body when it crosses the same point is? (ans : 1:7)
Answer:
Le H is the maximum height. Let h is the height above the ground at which both P.E and K.E are equal
∴ T e at the point p = K.E + P.E
⇒ K.E = mgH - mgh = mg [H - h]
∴ P.E / K.E = mgh/mg[Hh]
As K.E = P.E
∴ h / H - h = 1, H – h = 2h …(1)
When the body is projected upwards with velocity u it reaches a maximum height H
∴ H = u2/ 2g
When the body is projected vertically upwards with double the velocity then it reaches a maximum height H1
∴ H = (2u)2/2g = 4u2/2g = 4H………..(2)
∴ The ratio of P.E and K.E at the same point ‘P’ = P.E / K.E = mgh/mg[H1-h]
∴ P.E / K.E = h/4[2h]-h
= 1/7
Or, 1:7
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