Assertion : A particle is projected upwards with speed `upsilon` and it goes to a heigth `h`. If we double the speed then it will move to height `4h`.
Reason : In case of earth, acceleration due to gravity g varies as
`g prop(1)/(r^(2))` (for `r ge R`)

Options

A. If both Assertion and Reason are true and the Reason is correct explanation of the Assertion.
B. If both Assertion and Reason are true but Reason is not the correct explantion of Assertion.
C. If Assertion is true, but the Reason is false.
D. If Assertion is false but the Reason is true.

Explanation

To solve the given problem, we will analyze both the assertion and the reason step by step.

### Step 1: Understanding the Assertion
The assertion states that if a particle is projected upwards with speed \( v \) and reaches a height \( h \), then doubling the speed to \( 2v \) will allow it to reach a height of \( 4h \).

**Formula for Maximum Height:**
The maximum height \( h \) reached by a projectile is given by the formula:
\[ h = \frac{u^2}{2g} \]
where \( u \) is the initial velocity and \( g \) is the acceleration due to gravity.

### Step 2: Calculate Height for Initial Speed \( v \)
Using the formula for maximum height:
\[ h = \frac{v^2}{2g} \]

### Step 3: Calculate Height for Doubled Speed \( 2v \)
Now, if we double the speed to \( 2v \):
\[ h' = \frac{(2v)^2}{2g} = \frac{4v^2}{2g} = \frac{2v^2}{g} \]

### Step 4: Relate New Height \( h' \) to Original Height \( h \)
From the original height \( h = \frac{v^2}{2g} \), we can express \( v^2 \) in terms of \( h \):
\[ v^2 = 2gh \]
Substituting this into the equation for \( h' \):
\[ h' = \frac{2(2gh)}{g} = 4h \]
Thus, the assertion is correct.

### Step 5: Understanding the Reason
The reason states that the acceleration due to gravity \( g \) varies as \( g \propto \frac{1}{r^2} \) for \( r \geq R \), where \( R \) is the radius of the Earth.

### Step 6: Analyzing the Variation of Gravity
As a particle moves to a height \( h \) above the Earth's surface, the distance from the center of the Earth becomes \( r = R + h \). The formula for gravity at height \( h \) is:
\[ g' = \frac{g}{(1 + \frac{h}{R})^2} \]
This shows that \( g' \) decreases as \( h \) increases, confirming that \( g \) varies with \( r \).

### Conclusion
- The assertion is **correct**: Doubling the speed results in quadrupling the height.
- The reason is also **correct**: Gravity does vary with distance from the center of the Earth.

### Final Answer
Both the assertion and the reason are true.