Let escape velocity od a body a kept surface of a planet is u , If it is projected at a speed of 200 % more than the escape speed , then its speed in interstellar space will be

Options

A. u
B. `sqrt(3)` u
C. `2u`
D. `2sqrt(2)u`

Answer

To solve the problem, we need to determine the speed of a body in interstellar space when it is projected at a speed that is 200% more than the escape velocity of a planet. Let's break this down step by step.

### Step 1: Understand the given information
- Let the escape velocity of the planet be \( u \).
- The body is projected at a speed that is 200% more than the escape velocity.

### Step 2: Calculate the projected speed
- If the escape velocity is \( u \), then 200% of \( u \) is:
\[ 200\% \text{ of } u = 2u \]
- Therefore, the total speed at which the body is projected is:
\[ \text{Projected speed} = u + 2u = 3u \]

### Step 3: Use the formula for interstellar speed
- The formula for the interstellar speed \( V_{in} \) is given by:
\[ V_{in}^2 = V^2 - V_{escape}^2 \]
where \( V \) is the projected speed and \( V_{escape} \) is the escape velocity.

### Step 4: Substitute the values into the formula
- We have:
\[ V = 3u \quad \text{and} \quad V_{escape} = u \]
- Now substituting these values into the formula:
\[ V_{in}^2 = (3u)^2 - (u)^2 \]
\[ V_{in}^2 = 9u^2 - u^2 = 8u^2 \]

### Step 5: Calculate \( V_{in} \)
- Taking the square root of both sides gives:
\[ V_{in} = \sqrt{8u^2} = 2\sqrt{2}u \]

### Final Answer
Thus, the speed of the body in interstellar space will be:
\[ V_{in} = 2\sqrt{2}u \]

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