Question

A body moving through air at high speed v experiences a retarding force F given by F=k A D v^x, where A is the surface area of a body, D is density of air and k is dimensionless constant. Deduce the value of x.

Answer 1

Answer : the below pic will help you out with the understanding ans

Answer 2

Answer:

The value of x is 2.

Explanation:

Given a body moving through air a speed $v$.

The body experiences a retarding force, $F=k A D v^{x}$

where A is the surface area of a body, D is density of air and k is dimensionless constant.

Writing the dimensional formulae for all the quantities,

Force, $F=[M^1L^1T^{-2}]$

Area, $A=[M^0L^2T^0]$

Density, $D=[M^1L^{-3}T^0]$

Velocity, $v=[M^0L^1T^{-1}]$

Substituting the dimensions in the given formula of force, as k is a eliminating it and equating,

$[M^1L^1T^{-2}]=[M^0L^2T^0][M^1L^{-3}T^0] [M^0L^1T^{-1}]^{x}$

$[M^1L^1T^{-2}]=[M^1L^{2-3+x}T^{-x}]$

$[M^1L^1T^{-2}] = [M^1L^{-1+x} T^{-x}]$

Equating the powers on both sides of the equation,

From the powers of L,  $1= -1 +x\implies x=2$  

or  from the powers of T, $-2 = -x\implies x=2$

Therefore, the value of x is 2.

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