If the distance between sun and earth is increased by 3 times, then attraction between two will

Options

A. Remains constant
B. Decrease by `63%`
C. Increase by `63%`
D. Decreases `89%`

Explanation

To solve the problem of how the gravitational attraction between the Sun and the Earth changes when the distance between them is increased by three times, we can follow these steps:

### Step 1: Understand the Gravitational Force Formula
The gravitational force \( F \) between two masses \( M \) (mass of the Sun) and \( m \) (mass of the Earth) separated by a distance \( r \) is given by Newton's law of gravitation:
\[ F = \frac{G M m}{r^2} \]
where \( G \) is the gravitational constant.

### Step 2: Calculate the Initial Force
Let the initial distance between the Sun and Earth be \( r \). The initial gravitational force \( F \) can be expressed as:
\[ F = \frac{G M m}{r^2} \]

### Step 3: Determine the New Distance
If the distance between the Sun and Earth is increased by three times, the new distance becomes:
\[ d = 3r \]

### Step 4: Calculate the New Force
Using the new distance \( d = 3r \), the new gravitational force \( F' \) can be calculated as:
\[ F' = \frac{G M m}{(3r)^2} = \frac{G M m}{9r^2} \]

### Step 5: Relate the New Force to the Initial Force
We can express the new force \( F' \) in terms of the initial force \( F \):
\[ F' = \frac{1}{9} \cdot \frac{G M m}{r^2} = \frac{F}{9} \]

### Step 6: Analyze the Change in Force
This shows that the new gravitational force \( F' \) is \( \frac{1}{9} \) of the initial force \( F \). Therefore, the attraction has decreased.

### Step 7: Calculate the Percentage Decrease
To find the percentage decrease in gravitational force, we can use the formula:
\[ \text{Percentage Decrease} = \frac{F - F'}{F} \times 100 \]
Substituting the values:
\[ \text{Percentage Decrease} = \frac{F - \frac{F}{9}}{F} \times 100 = \frac{F \left(1 - \frac{1}{9}\right)}{F} \times 100 = \frac{8F/9}{F} \times 100 = \frac{8}{9} \times 100 \approx 88.89\% \]

### Conclusion
Thus, the gravitational attraction between the Sun and Earth decreases by approximately 89%.

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