1.

The depth x to which a bullet penetrates a human body depends on (i) coeffeicint of elasticity, `eta` and (ii) KE `(E_k)` of the bullet, By the method of dimensions, show that `x prop ((E_k)/(eta))^(1//3)`

Answer» Let `x prop E_k^a eta^b,`
where, a, b are the dimensions `x = K E_k^a eta^b ….(i)`
where K is dimansionless constant fo
proportionality.
Writing the dimensions in (i), we get
`[M^0L^1 T^0] = (ML^2 T^(-2))^a (ML^(-1) T^(-2))^b`
`=M^(a +b) L^(2a -b) T^(-2a -2b)`
Applying the principle of homogeneity of dimensions, we get
` a +b =0 .....(ii)`
`2a - b =1 ....(iii)`
`-2a -2b =0 ....(iv)`
Add (ii) and (iii) : `3a =1, a = 1//3`
Form (ii), `b =-a - (1)/(3)`.
From (i),` x =K((E_k)/(eta))^(1//3)`
Henc, `x prop((E_k)/(eta))^(1//3)`


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