A particle of mass `2 kg` moves in the `xy` plane under the action of a constant force `vec(F)` where `vec(F)=hat(i)-hat(j)`. Initially the velocity of the particle is `2hat(j)`. The velocity of the particle at time `t` is

Options

A. `(1)/(2)(t+4)hat(i)-(1)/(2)that(j)`
B. `t(hat(i)-hat(j))`
C. `(1)/(2)t(hat(i)-hat(j))`
D. `(1)/(2)that(i)+(1)/(2)(t+4)hat(j)`

Explanation

To find the velocity of a particle of mass 2 kg moving under the action of a constant force \(\vec{F} = \hat{i} - \hat{j}\) at time \(t\), we can follow these steps:

### Step 1: Identify the given quantities
- Mass of the particle, \(m = 2 \, \text{kg}\)
- Force acting on the particle, \(\vec{F} = \hat{i} - \hat{j}\)
- Initial velocity of the particle, \(\vec{u} = 2\hat{j}\)

### Step 2: Calculate the acceleration
Using Newton's second law, the acceleration \(\vec{a}\) can be calculated using the formula:
\[ \vec{a} = \frac{\vec{F}}{m} \]
Substituting the values:
\[ \vec{a} = \frac{\hat{i} - \hat{j}}{2} = \frac{1}{2}\hat{i} - \frac{1}{2}\hat{j} \]

### Step 3: Use the equation of motion to find the velocity at time \(t\)
The equation of motion that relates initial velocity, acceleration, and final velocity is:
\[ \vec{v} = \vec{u} + \vec{a} t \]
Substituting the values we have:
\[ \vec{v} = 2\hat{j} + \left(\frac{1}{2}\hat{i} - \frac{1}{2}\hat{j}\right)t \]
Distributing \(t\) into the acceleration:
\[ \vec{v} = 2\hat{j} + \frac{1}{2}t\hat{i} - \frac{1}{2}t\hat{j} \]

### Step 4: Combine the terms
Now, we can combine the terms involving \(\hat{j}\):
\[ \vec{v} = \frac{1}{2}t\hat{i} + \left(2 - \frac{1}{2}t\right)\hat{j} \]

### Final Result
Thus, the velocity of the particle at time \(t\) is:
\[ \vec{v} = \frac{1}{2}t\hat{i} + \left(2 - \frac{1}{2}t\right)\hat{j} \]