What is the linear velocity of a body on the surface of the earth at the equator ? Given the radius of the earth is 6400 km . Period of rotation of the earth = 24 hours.

Options

A. `465 "m s"^(-1)`
B. `165 "m s"^(-1)`
C. `665 "m s"^(-1)`
D. `410 "m s"^(-1)`

Answer

To find the linear velocity of a body on the surface of the Earth at the equator, we can follow these steps:

### Step 1: Understand the concept of linear velocity
Linear velocity (V) is defined as the distance traveled per unit time. For an object moving in a circular path, it can be calculated using the formula:
\[ V = \frac{D}{T} \]
where \( D \) is the distance traveled in one complete rotation and \( T \) is the time taken for that rotation.

### Step 2: Calculate the distance traveled in one complete rotation
The distance \( D \) traveled in one complete rotation around the Earth at the equator can be calculated using the formula for the circumference of a circle:
\[ D = 2 \pi r \]
where \( r \) is the radius of the Earth. Given that the radius of the Earth is 6400 km, we first convert this to meters:
\[ r = 6400 \text{ km} = 6400 \times 10^3 \text{ m} \]

Now, substituting the value of \( r \) into the equation for \( D \):
\[ D = 2 \pi (6400 \times 10^3) \]

### Step 3: Calculate the time taken for one complete rotation
The period of rotation of the Earth is given as 24 hours. To convert this into seconds (since we want the velocity in meters per second), we use:
\[ T = 24 \text{ hours} \times 60 \text{ minutes/hour} \times 60 \text{ seconds/minute} \]
\[ T = 24 \times 60 \times 60 \text{ seconds} \]

### Step 4: Substitute the values into the velocity formula
Now we can substitute the values of \( D \) and \( T \) into the linear velocity formula:
\[ V = \frac{D}{T} \]
\[ V = \frac{2 \pi (6400 \times 10^3)}{24 \times 60 \times 60} \]

### Step 5: Calculate the final value of linear velocity
Now we can compute the value:
1. Calculate \( D \):
\[ D = 2 \pi (6400 \times 10^3) \approx 4021238.34 \text{ m} \]
2. Calculate \( T \):
\[ T = 24 \times 60 \times 60 = 86400 \text{ seconds} \]
3. Now calculate \( V \):
\[ V = \frac{4021238.34}{86400} \approx 46.56 \text{ m/s} \]

### Final Answer
The linear velocity of a body on the surface of the Earth at the equator is approximately:
\[ V \approx 465.42 \text{ m/s} \]

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