Question

The diameter of ball y is double that of x. The ratio of their terminal velocities inside water will be:
A) 1:4
B) 4:1
C) 1:2
D) 2:1

Answer 1

Hint
Here, we will be using the formula of the terminal velocity $V$ of the body of radius $r$, density $\rho $falling through a medium of density${\rho _o}$ ​ is given by $V = \dfrac{{2{r^2}\left( {\rho - {\rho _o}} \right)g}}{{9\eta }}$ where $\eta $ is the coefficient of viscosity of medium.

Complete step-by-step solution
Given, two balls $x$ and $y$ are flowing in water which have attained terminal velocity.
The diameter of ball $y$ is double that of ball $x$. The radius is just half of the diameter so it implies that the radius of ball $y$ is also double that of $x$.
Therefore, radius of ball $y$ is also double that of $x$ i.e. ${r_y} = 2{r_x}$
As we can clearly see from formula $V\infty {r^2}$
Therefore, the terminal velocity of ball x and y inside water will be $\dfrac{{{V_x}}}{{{V_y}}} = {\left( {\dfrac{{{r_x}}}{{{r_y}}}} \right)^2}$
Now, as given${r_y} = 2{r_x}$. Put this value in above equation, we get
$ \Rightarrow \dfrac{{{V_x}}}{{{V_y}}} = {\left( {\dfrac{{{r_x}}}{{2{r_x}}}} \right)^2}$
$ \Rightarrow \dfrac{{{V_x}}}{{{V_y}}} = \left( {\dfrac{{{r_x}^2}}{{4{r_x}^2}}} \right)$
 $ \Rightarrow \dfrac{{{V_x}}}{{{V_y}}} = \dfrac{1}{4}$ or ${V_x}:{V_y}$= 1:4
Hence the ratio of ${V_y}:{V_x}$ is 4: 1.
The ratio of terminal velocities of $y$ and $x$ inside water will be 4:1.
Thus, Option (B) is correct.

Note
In these types of questions, students may go wrong in identifying the correct ratio according to the question. One can do mistake by taking ratio of ${V_x}:{V_y}$ i.e. 1:4 and which is incorrect but according to question we need to write in form ${V_y}:{V_x}$ i.e. 4: 1 and which is correct answer.