Question

The depth x to which a bullet penetrates the skin depends upon the coefficient of elasticity and kinetic energy. By dimensions establish a relation amongst the physical quantities.

Answer 1

Depth x

x = [ L ] = e ᵃ (ke) ᵇ

where e = elasticity coeff.
         ke = kinetic energy

e ᵃ = [ M ]ᵃ [ L ] ⁻ᵃ [ T ] ⁻²ᵃ

(ke) ᵇ = [ M ]ᵇ [ L ] ²ᵇ [ T ] ⁻²ᵇ

a + b = 0

-a + 2b = 1

Solving these we get

3b = 1

b = 1/3

So, a = -1/3

Therefore, we have

$x = e^{-1/3} (ke)^{1/3}$

Answer 2

coefficient of elasticity is Young's modulus /bulk modulus
we know, dimension of Young's modulus/bulk modulus= dimension of pressure
∴ dimension of coefficient of elasticity = dimension of pressure
= [ML⁻¹T⁻²]
dimension of kinetic energy = [ML²T⁻²]
dimension of depth , x = [L]

Let relation between, depth, Coefficient of elasticity and kinetic energy is given by depth =k (Coefficient of elasticity)ᵃ (kinetic energy)ᵇ , here k is constant
∴ [L] = k[ML⁻¹T⁻²]ᵃ [ML²T⁻²]ᵇ = [Mᵃ⁺ᵇ ] [L⁻ᵃ⁺²ᵇ] [T⁻²ᵃ ⁻²ᵇ]
Compare both sides,
a + b = 0
-a + 2b = 1
-2(a + b) = 0⇒a + b = 0
Solve these equations ,
b = 1/3 and a = -1/3

∴$\bold{depth=(coefficient\:of\: elasticity)^{-1/3}(kinetic \:energy)^{1/3}}$