A rain drop with radus 1.5 mm falls from a cloud at a height 1200 m from ground . The density of water is 1000 `kg//m^(3)` and density of air is `1.2 kg//m^(3)`. Assume the drop was spherical throughout the fall and there is no air drag. The impact speed of the drop will be :

Options

A. 27 km/h
B. 550 km/h
C. Zero
D. 129 km/h

Answer

To solve the problem step by step, we will calculate the impact speed of a raindrop falling from a height of 1200 m, considering the given parameters.

### Step 1: Understand the Problem
We have a raindrop with a radius of 1.5 mm falling from a height of 1200 m. The density of water is 1000 kg/m³, and the density of air is 1.2 kg/m³. We need to find the impact speed of the raindrop when it reaches the ground, assuming no air drag.

### Step 2: Calculate the Volume of the Raindrop
The raindrop is spherical, so we can calculate its volume using the formula for the volume of a sphere:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the raindrop.

Given \( r = 1.5 \text{ mm} = 0.0015 \text{ m} \):
\[ V = \frac{4}{3} \pi (0.0015)^3 \]
Calculating this gives:
\[ V \approx 1.4137 \times 10^{-8} \text{ m}^3 \]

### Step 3: Calculate the Mass of the Raindrop
Using the density of water, we can find the mass of the raindrop:
\[ M = \rho_w \times V \]
where \( \rho_w = 1000 \text{ kg/m}^3 \):
\[ M = 1000 \times 1.4137 \times 10^{-8} \approx 1.4137 \times 10^{-5} \text{ kg} \]

### Step 4: Calculate the Upward Thrust (Buoyant Force)
The upward thrust (buoyant force) can be calculated using Archimedes' principle:
\[ F_{up} = \rho_a \times V \times g \]
where \( \rho_a = 1.2 \text{ kg/m}^3 \) and \( g \approx 9.81 \text{ m/s}^2 \):
\[ F_{up} = 1.2 \times 1.4137 \times 10^{-8} \times 9.81 \approx 1.67 \times 10^{-7} \text{ N} \]

### Step 5: Calculate the Net Force Acting on the Raindrop
The net force acting on the raindrop is given by:
\[ F_{net} = mg - F_{up} \]
Calculating \( mg \):
\[ mg = 1.4137 \times 10^{-5} \times 9.81 \approx 1.39 \times 10^{-4} \text{ N} \]
Now, substituting the values:
\[ F_{net} = 1.39 \times 10^{-4} - 1.67 \times 10^{-7} \approx 1.39 \times 10^{-4} \text{ N} \]

### Step 6: Calculate the Acceleration of the Raindrop
Using Newton's second law, we can find the acceleration:
\[ a = \frac{F_{net}}{M} \]
\[ a = \frac{1.39 \times 10^{-4}}{1.4137 \times 10^{-5}} \approx 9.81 \text{ m/s}^2 \]

### Step 7: Calculate the Final Velocity Using Kinematics
Using the kinematic equation:
\[ v^2 = u^2 + 2as \]
where \( u = 0 \) (initial velocity), \( a = 9.81 \text{ m/s}^2 \), and \( s = 1200 \text{ m} \):
\[ v^2 = 0 + 2 \times 9.81 \times 1200 \]
Calculating this gives:
\[ v^2 = 23544 \implies v \approx 153.5 \text{ m/s} \]

### Step 8: Convert to Kilometers per Hour
To convert the speed from m/s to km/h:
\[ v_{km/h} = v_{m/s} \times \frac{3600}{1000} = 153.5 \times 3.6 \approx 552.6 \text{ km/h} \]

### Final Answer
The impact speed of the raindrop is approximately **553 km/h**.