Question

Terminal velocity of Mars rover depends upon
P = gas density of atmosphere of mars
A = frontal area
W = weight of falling object
What may be the possible relation between terminal velocity $v$ and given quantity.

• A. $v=\frac{w}{pA}$
• B. $v=\sqrt{\frac{w}{pA}}$
• C. $v=\sqrt{\frac{pA}{w}}$
• D. $v=\sqrt{\frac{p{A}^3}{w}}$

Answer 1

The correct option is B $v=\sqrt{\frac{w}{pA}}$

$v=p^x{A}^y{w}^z$
(x, y, z $\to$ unknown)
$\Rightarrow L{T}^{-1}=\left(M{L}^{-3}{)}^x\right(L^2{)}^y(ML{T}^{-2}{)}^z$
$M^o{L}^1{T}^{-1}=M^{x+z}{L}^{-3x+2y+z}{T}^{-2z}$
Power of M $\to$ x + z =0
Power of T $\to$ -2z = -1
Power of L $\to$ -3x + 2y + z =1

$x=-z=\frac{-1}{2}$
$z=\frac{1}{2}x=\frac{-1}{2}$
$-3x+2y+z=1$
$\frac{3}{2}+2y+\frac{1}{2}=12y=1-\frac{4}{2}=-1$
$y=\frac{-1}{2}$

$v=p^{\frac{-1}{2}}.A^{\frac{-1}{2}}.w^{\frac{1}{2}}$
$v=\sqrt{\frac{w}{pA}}$