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Question
Derive the formula of the kinetic energy of a particle having mass ‘m’ and velocity ‘v’, using dimensional analysis.
Solution
The kinetic energy of a body depends upon mass (m) and velocity (v) of the body.
Let K.E. ∝ mx vy
∴ K.E. = kmx vy ...(1)
where,
k = dimensionless constant of proportionality.
Taking dimensions on both sides of equation (1),
[L2M1T-2]
= [L0 M1 T0]x [L1 M0 T-1]y
= [L0 Mx T0] [Ly M0 T-y]
= `["L"^(0+y) " M"^(x + 0) " T"^(0- y)]`
[L2 M1 T-2] = [Ly Mx T-y] ...(2)
Equating dimensions of L, M, T on both sides of equation (2),
x = 1 and y = 2
Substituting x, y in equation (1), we have
K.E. = kmv2
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