To solve the problem of finding the impact speed of a raindrop falling from a height of 1200 m, we will follow these steps:
### Step 1: Calculate the Volume of the Raindrop
The raindrop is spherical, and its volume \( V \) can be calculated using the formula for the volume of a sphere:
\[
V = \frac{4}{3} \pi r^3
\]
Given the radius \( r = 1.5 \, \text{mm} = 0.0015 \, \text{m} \):
\[
V = \frac{4}{3} \pi (0.0015)^3 = \frac{4}{3} \pi (3.375 \times 10^{-8}) \approx 1.4137 \times 10^{-7} \, \text{m}^3
\]
### Step 2: Calculate the Mass of the Raindrop
Using the density of water \( \rho_w = 1000 \, \text{kg/m}^3 \), the mass \( m \) of the raindrop can be calculated as:
\[
m = \rho_w \cdot V = 1000 \cdot 1.4137 \times 10^{-7} \approx 1.4137 \times 10^{-4} \, \text{kg}
\]
### Step 3: Calculate the Weight of the Raindrop
The weight \( W \) of the raindrop is given by:
\[
W = m \cdot g
\]
Where \( g \approx 10 \, \text{m/s}^2 \):
\[
W = 1.4137 \times 10^{-4} \cdot 10 \approx 1.4137 \times 10^{-3} \, \text{N}
\]
### Step 4: Calculate the Upward Thrust (Buoyant Force)
The upward thrust (buoyant force) \( F_b \) can be calculated using the density of air \( \rho_a = 1.2 \, \text{kg/m}^3 \):
\[
F_b = \rho_a \cdot V \cdot g = 1.2 \cdot 1.4137 \times 10^{-7} \cdot 10 \approx 1.6964 \times 10^{-6} \, \text{N}
\]
### Step 5: Calculate the Net Force Acting on the Raindrop
The net force \( F_{net} \) acting on the raindrop is the difference between its weight and the buoyant force:
\[
F_{net} = W - F_b = 1.4137 \times 10^{-3} - 1.6964 \times 10^{-6} \approx 1.4137 \times 10^{-3} \, \text{N}
\]
### Step 6: Calculate the Acceleration of the Raindrop
Using Newton's second law, the acceleration \( a \) of the raindrop can be calculated as:
\[
a = \frac{F_{net}}{m} = \frac{1.4137 \times 10^{-3}}{1.4137 \times 10^{-4}} \approx 10 \, \text{m/s}^2
\]
### Step 7: Use the Kinematic Equation to Find the Final Velocity
Using the kinematic equation \( v^2 = u^2 + 2as \), where \( u = 0 \) (initial velocity), \( a = 10 \, \text{m/s}^2 \), and \( s = 1200 \, \text{m} \):
\[
v^2 = 0 + 2 \cdot 10 \cdot 1200
\]
\[
v^2 = 24000
\]
\[
v = \sqrt{24000} \approx 154.92 \, \text{m/s}
\]
### Step 8: Convert the Velocity to km/h
To convert from m/s to km/h, we multiply by \( \frac{18}{5} \):
\[
v_{km/h} = 154.92 \cdot \frac{18}{5} \approx 557.0 \, \text{km/h}
\]
### Final Answer
The impact speed of the raindrop will be approximately **557 km/h**.
---
