To calculate the velocity of a particle dropped down a deep hole at a depth of 1 km from the surface of the Earth, we can use the concept of gravitational acceleration and energy conservation. Here’s a step-by-step solution:
### Step 1: Understand the Problem
We need to find the velocity of a particle at a depth of 1 km from the surface of the Earth. We are given:
- Gravitational acceleration at the surface, \( g = 10 \, \text{m/s}^2 \)
- Radius of the Earth, \( R = 6400 \, \text{km} = 6400000 \, \text{m} \)
### Step 2: Calculate the Depth
The depth \( d \) from the surface is given as 1 km, which is:
\[ d = 1 \, \text{km} = 1000 \, \text{m} \]
### Step 3: Calculate the Distance from the Center of the Earth
The distance from the center of the Earth at a depth of 1 km can be calculated as:
\[ r = R - d = 6400000 \, \text{m} - 1000 \, \text{m} = 6399000 \, \text{m} \]
### Step 4: Calculate the Gravitational Acceleration at Depth
The gravitational acceleration \( g' \) at a depth \( d \) can be calculated using the formula:
\[ g' = g \left( \frac{r}{R} \right) \]
Substituting the values:
\[ g' = 10 \, \text{m/s}^2 \left( \frac{6399000}{6400000} \right) \approx 10 \, \text{m/s}^2 \]
### Step 5: Use Energy Conservation to Find Velocity
When the particle is dropped, it converts its potential energy into kinetic energy. The potential energy lost equals the kinetic energy gained. The potential energy change can be approximated as:
\[ \Delta PE = m g d \]
The kinetic energy at depth \( d \) is given by:
\[ KE = \frac{1}{2} mv^2 \]
Setting these equal gives:
\[ m g d = \frac{1}{2} mv^2 \]
The mass \( m \) cancels out:
\[ g d = \frac{1}{2} v^2 \]
Rearranging gives:
\[ v^2 = 2 g d \]
Substituting the values:
\[ v^2 = 2 \times 10 \, \text{m/s}^2 \times 1000 \, \text{m} \]
\[ v^2 = 20000 \]
Taking the square root:
\[ v = \sqrt{20000} \]
\[ v \approx 141.42 \, \text{m/s} \]
### Step 6: Convert to km/s
To convert the velocity from m/s to km/s:
\[ v \approx \frac{141.42}{1000} \approx 0.14142 \, \text{km/s} \]
### Final Answer
The velocity of the particle at a depth of 1 km from the surface of the Earth is approximately:
\[ v \approx 0.141 \, \text{km/s} \]
