Two bodies begin a free fall from the same height at a time interval of Ns. If vertical separation between the two bodies is Lm after n seconds from the start of the first body, then n is equal to

Options

A. `sqrt(nN)`
B. `L/(gN)`
C. `L/(gN) + N/2`
D. `L/(gN) - N/4`

Answer

To solve the problem, we need to analyze the motion of the two bodies in free fall and their vertical separation after a given time. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the motion of the first body
The first body starts falling from rest at time \( t = 0 \). The distance fallen by the first body after \( n \) seconds can be calculated using the equation of motion:
\[ x_1 = \frac{1}{2} g n^2 \]
where \( g \) is the acceleration due to gravity.

### Step 2: Understand the motion of the second body
The second body starts falling \( N \) seconds after the first body. Therefore, the time of fall for the second body after \( n \) seconds from the start of the first body is \( (n - N) \) seconds. The distance fallen by the second body can be expressed as:
\[ x_2 = \frac{1}{2} g (n - N)^2 \]

### Step 3: Calculate the vertical separation between the two bodies
The vertical separation \( L \) between the two bodies after \( n \) seconds is given by:
\[ L = x_1 - x_2 \]
Substituting the expressions for \( x_1 \) and \( x_2 \):
\[ L = \frac{1}{2} g n^2 - \frac{1}{2} g (n - N)^2 \]

### Step 4: Simplify the expression for \( L \)
Expanding the second term:
\[ (n - N)^2 = n^2 - 2nN + N^2 \]
Thus,
\[ L = \frac{1}{2} g n^2 - \frac{1}{2} g (n^2 - 2nN + N^2) \]
This simplifies to:
\[ L = \frac{1}{2} g (2nN - N^2) = g n N - \frac{1}{2} g N^2 \]

### Step 5: Rearranging the equation
From the above equation, we can express \( n \):
\[ L = g n N - \frac{1}{2} g N^2 \]
Rearranging gives:
\[ g n N = L + \frac{1}{2} g N^2 \]
Thus,
\[ n = \frac{L + \frac{1}{2} g N^2}{g N} \]

### Step 6: Solve for \( n \)
To find \( n \) in terms of \( L \) and \( N \):
\[ n = \frac{L}{g N} + \frac{N}{2} \]

### Conclusion
From the problem statement, if the vertical separation \( L \) is given, we can find \( n \) using the derived formula. If we set \( L = 1 \) m and \( N = 1 \) s, we can substitute these values to find \( n \).

### Final Answer
After simplification, we find that \( n \) is equal to \( 2 \) seconds.

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