To solve the problem, we will follow these steps:
### Step 1: Identify the forces acting on the block
The block has a mass of \( m = 4 \, \text{kg} \). It is subjected to:
- An initial velocity \( \vec{u} = 3 \hat{i} \, \text{m/s} \).
- A force \( \vec{F} = -2 \hat{i} \, \text{N} \) acting in the opposite direction.
- A frictional force due to the rough surface.
### Step 2: Calculate the normal force and the frictional force
The normal force \( N \) acting on the block is equal to its weight since it is on a horizontal surface:
\[
N = mg = 4 \, \text{kg} \times 10 \, \text{m/s}^2 = 40 \, \text{N}
\]
The frictional force \( f \) is given by:
\[
f = \mu N = 0.1 \times 40 \, \text{N} = 4 \, \text{N}
\]
Since the frictional force acts in the opposite direction to the motion, it will also be negative.
### Step 3: Calculate the net force acting on the block
The total force acting on the block is the applied force plus the frictional force:
\[
F_{\text{net}} = -2 \, \text{N} - 4 \, \text{N} = -6 \, \text{N}
\]
### Step 4: Calculate the acceleration of the block
Using Newton's second law \( F = ma \):
\[
-6 \, \text{N} = 4 \, \text{kg} \cdot a
\]
Solving for \( a \):
\[
a = \frac{-6 \, \text{N}}{4 \, \text{kg}} = -\frac{3}{2} \, \text{m/s}^2
\]
### Step 5: Determine the time until the block comes to rest
Using the equation of motion \( v = u + at \), where \( v = 0 \) (final velocity when the block comes to rest):
\[
0 = 3 \, \text{m/s} + \left(-\frac{3}{2} \, \text{m/s}^2\right) t
\]
Rearranging gives:
\[
\frac{3}{2} t = 3 \quad \Rightarrow \quad t = \frac{3 \times 2}{3} = 2 \, \text{s}
\]
### Step 6: Calculate the displacement during the first 2 seconds
Using the equation for displacement \( s = ut + \frac{1}{2}at^2 \):
\[
s = (3 \, \text{m/s})(2 \, \text{s}) + \frac{1}{2}\left(-\frac{3}{2} \, \text{m/s}^2\right)(2 \, \text{s})^2
\]
Calculating each part:
\[
s = 6 \, \text{m} - \frac{1}{2} \times \frac{3}{2} \times 4 = 6 \, \text{m} - 3 \, \text{m} = 3 \, \text{m}
\]
### Final Answer
The displacement of the block in the first 5 seconds is:
\[
\boxed{3 \hat{i} \, \text{m}}
\]
