Two bodies are thrown verically upward, with the same initially velocity of `98 m//s`, but the second one is thrown `4` sec. after first is thrown. If after `4n` seconds after the first particle is thrown, both of them meet then value of `n` is ?

Explanation

To solve the problem, we need to analyze the motion of two bodies thrown vertically upward with the same initial velocity of \( 98 \, \text{m/s} \). The second body is thrown \( 4 \) seconds after the first body. We want to find the value of \( n \) such that both bodies meet after \( 4n \) seconds from the time the first body is thrown.

### Step-by-Step Solution:

1. **Define the Variables**:
- Let \( t \) be the time in seconds after the first body is thrown.
- The first body is thrown at \( t = 0 \).
- The second body is thrown at \( t = 4 \) seconds.

2. **Equation of Motion**:
- The height \( h_1 \) of the first body after \( t \) seconds can be given by the equation:
\[ h_1 = ut - \frac{1}{2}gt^2 \]
where \( u = 98 \, \text{m/s} \) and \( g = 10 \, \text{m/s}^2 \) (acceleration due to gravity).
- Thus, substituting the values:
\[ h_1 = 98t - \frac{1}{2} \cdot 10 \cdot t^2 = 98t - 5t^2 \quad \text{(Equation 1)} \]

3. **Height of the Second Body**:
- The height \( h_2 \) of the second body after \( t - 4 \) seconds (since it is thrown 4 seconds later) is given by:
\[ h_2 = u(t - 4) - \frac{1}{2}g(t - 4)^2 \]
- Substituting the values:
\[ h_2 = 98(t - 4) - 5(t - 4)^2 \]
- Expanding this:
\[ h_2 = 98t - 392 - 5(t^2 - 8t + 16) = 98t - 392 - 5t^2 + 40t - 80 \]
\[ h_2 = 98t + 40t - 5t^2 - 472 = 138t - 5t^2 - 472 \quad \text{(Equation 2)} \]

4. **Setting the Heights Equal**:
- For the two bodies to meet, their heights must be equal:
\[ h_1 = h_2 \]
- Therefore, we set Equation 1 equal to Equation 2:
\[ 98t - 5t^2 = 138t - 5t^2 - 472 \]

5. **Simplifying the Equation**:
- Cancel \( -5t^2 \) from both sides:
\[ 98t = 138t - 472 \]
- Rearranging gives:
\[ 138t - 98t = 472 \]
\[ 40t = 472 \]
\[ t = \frac{472}{40} = 11.8 \, \text{seconds} \]

6. **Finding \( n \)**:
- We know that the meeting time \( t \) is also expressed as \( 4n \):
\[ 4n = 11.8 \]
- Solving for \( n \):
\[ n = \frac{11.8}{4} = 2.95 \]

7. **Final Answer**:
- Since \( n \) must be a whole number, we round \( n \) to the nearest whole number:
\[ n = 3 \]

### Final Result:
The value of \( n \) is \( 3 \).