Question

If two rain drops of radii $r_1$ and $r_2$ reach the ground with terminal velocities $v_1$ and $v_2$ and linear momenta $p$ and $32p$ respectively, then $r_1 : r_2 =$

• A. 1 : 16
• B. 1 : 2
• C. 2 : 1
• D. 16 : 1

Hint:

Remember the proportionality relations for spherical objects in viscous fluids: terminal velocity $v_t \propto r^2$ and mass $m \propto r^3$. These are crucial for relating momentum to radius.

Answer 1

The Correct Option is B

Step 1: Relate terminal velocity to radius.
For a small spherical object falling through a viscous medium (like air), the terminal velocity ($v_t$) is given by Stokes' Law (assuming density of fluid is negligible compared to density of object): \[ v_t = \frac{2 r^2 g (\rho - \sigma)}{9 \eta} \] Where $r$ is the radius, $g$ is acceleration due to gravity, $\rho$ is the density of the raindrop, $\sigma$ is the density of air, and $\eta$ is the viscosity of air. Since $g$, $\rho$, $\sigma$, and $\eta$ are constants for this scenario: \[ v_t \propto r^2 \] So, for the two raindrops: \[ \frac{v_1}{v_2} = \frac{r_1^2}{r_2^2} \quad \cdots (1) \] Step 2: Relate linear momentum to mass and velocity.
Linear momentum ($p$) is given by $p = mv$.
The mass ($m$) of a spherical raindrop is $m = \frac{4}{3}\pi r^3 \rho$. Therefore, the momentum of a raindrop is:
\[ p \propto r^3 v \] For the two raindrops: \[ \frac{p_1}{p_2} = \frac{r_1^3 v_1}{r_2^3 v_2} \quad \cdots (2) \] Step 3: Substitute the velocity ratio into the momentum ratio.
We are given $p_1 = p$ and $p_2 = 32p$. Substitute $\frac{v_1}{v_2} = \frac{r_1^2}{r_2^2}$ from (1) into (2): \[ \frac{p}{32p} = \frac{r_1^3}{r_2^3} \times \frac{r_1^2}{r_2^2} \] \[ \frac{1}{32} = \frac{r_1^{3+2}}{r_2^{3+2}} = \frac{r_1^5}{r_2^5} \] \[ \frac{1}{32} = \left(\frac{r_1}{r_2}\right)^5 \] Step 4: Solve for the ratio of radii.
To find $\frac{r_1}{r_2}$, take the fifth root of both sides: \[ \frac{r_1}{r_2} = \left(\frac{1}{32}\right)^{1/5} = \frac{1^{1/5}}{32^{1/5}} = \frac{1}{2} \] So, $r_1 : r_2 = 1 : 2$. The final answer is $\boxed{\text{1 : 2}}$.