Question

Derive an expression of kinetic energy of a body of mass ‘m’ and moving with velocity ‘v’, using dimensional analysis.

Answer 1

Answer:

Suppose a stationery object of mass m moves because of an applied force.Let u be its initial velocity (here u = 0).Let the applied force be F. This generates an acceleration a in the object, and, after time t, the velocity of the becomes equal to v. The displacement during this time is s.The work done on the object, W=F×s

According to Newton's second law of motion,

F=ma ------------ (1)Similarly, using Newton's second equation of motion

However, as initial velocity is zero,u=0

s=ut + 1/2 at2

s=0 + 1/2 at2

s= 1/2 at2 ------------(2)

:. W=ma × 1/2 at2 -----------using equations (1) and(2)

W=1/2 m(at)2 -------------(3)

Using Newton's first equation of motion.............

v= u + at

:.v= 0 + at

:.v= at

:.v2 = (at)2 ----------(4)

:.W=1/2 mv2 -----------using equations (3) and (4)

The kinetic energy gained by an object is the amount of work done on the object.

:.K.E. = W

:.K.E. = 1/2 mv2

Hope it helps you