To solve the problem step by step, we will first find the acceleration of the particle and then calculate the displacement at \( t = 1 \, \text{s} \).
### Step 1: Find the acceleration
The velocity of the particle is given by:
\[
v = 2 \hat{i} + 2t \hat{j} \, \text{m/s}
\]
Acceleration \( a \) is defined as the rate of change of velocity with respect to time, which can be expressed mathematically as:
\[
a = \frac{dv}{dt}
\]
To find \( a \), we differentiate the velocity \( v \) with respect to time \( t \):
\[
\frac{dv}{dt} = \frac{d}{dt}(2 \hat{i} + 2t \hat{j}) = 0 \hat{i} + 2 \hat{j} = 2 \hat{j} \, \text{m/s}^2
\]
Thus, the acceleration of the particle is:
\[
a = 2 \hat{j} \, \text{m/s}^2
\]
### Step 2: Determine if we can apply \( v = u + at \)
In order to apply the equation \( v = u + at \), the acceleration must be constant. Since we found that the acceleration \( a = 2 \hat{j} \) is a constant value, we can indeed apply this equation.
### Step 3: Find the displacement at \( t = 1 \, \text{s} \)
Displacement \( s \) can be calculated using the integral of the velocity over time:
\[
s = \int v \, dt
\]
We need to evaluate this integral from \( t = 0 \) to \( t = 1 \):
\[
s = \int_0^1 (2 \hat{i} + 2t \hat{j}) \, dt
\]
Breaking this down, we can integrate each component separately:
\[
s = \int_0^1 2 \hat{i} \, dt + \int_0^1 2t \hat{j} \, dt
\]
Calculating the first integral:
\[
\int_0^1 2 \hat{i} \, dt = 2 \hat{i} \cdot [t]_0^1 = 2 \hat{i} \cdot (1 - 0) = 2 \hat{i}
\]
Calculating the second integral:
\[
\int_0^1 2t \hat{j} \, dt = 2 \hat{j} \cdot \left[\frac{t^2}{2}\right]_0^1 = 2 \hat{j} \cdot \left(\frac{1^2}{2} - 0\right) = 2 \hat{j} \cdot \frac{1}{2} = \hat{j}
\]
Combining both results, we find the total displacement:
\[
s = 2 \hat{i} + \hat{j} \, \text{m}
\]
### Final Answers
- Acceleration: \( 2 \hat{j} \, \text{m/s}^2 \)
- Displacement at \( t = 1 \, \text{s} \): \( 2 \hat{i} + \hat{j} \, \text{m} \)
